Misplaced Pages

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110-410: In kinematics , the speed (commonly referred to as v ) of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval; the instantaneous speed

220-416: A B x , a B y , a B z ) {\displaystyle \mathbf {a} _{B}=\left(a_{B_{x}},a_{B_{y}},a_{B_{z}}\right)} then the acceleration of point C relative to point B is the difference between their components: a C / B = a C − a B = (

330-460: A C x − a B x , a C y − a B y , a C z − a B z ) {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}=\left(a_{C_{x}}-a_{B_{x}},a_{C_{y}}-a_{B_{y}},a_{C_{z}}-a_{B_{z}}\right)} Alternatively, this same result could be obtained by computing

440-443: A | = | v ˙ | = d v d t . {\displaystyle |\mathbf {a} |=|{\dot {\mathbf {v} }}|={\frac {{\text{d}}v}{{\text{d}}t}}.} A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points. The position of one point A relative to another point B

550-541: A τ ) d τ = r 0 + v 0 t + 1 2 a t 2 . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\int _{0}^{t}\mathbf {v} (\tau )\,{\text{d}}\tau =\mathbf {r} _{0}+\int _{0}^{t}\left(\mathbf {v} _{0}+\mathbf {a} \tau \right){\text{d}}\tau =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}.} Additional relations between displacement, velocity, acceleration, and time can be derived. Since

660-496: A ⋅ x ) {\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})} where v = | v | etc. The above equations are valid for both Newtonian mechanics and special relativity . Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on

770-605: A ) ⋅ x = ( 2 a ) ⋅ ( u t + 1 2 a t 2 ) = 2 t ( a ⋅ u ) + a 2 t 2 = v 2 − u 2 {\displaystyle (2{\boldsymbol {a}})\cdot {\boldsymbol {x}}=(2{\boldsymbol {a}})\cdot ({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2})=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}} ∴ v 2 = u 2 + 2 (

880-447: A = d v d t . {\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.} From there, velocity is expressed as the area under an a ( t ) acceleration vs. time graph. As above, this is done using the concept of the integral: v = ∫ a   d t . {\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.} In

990-410: A r = − v θ , a θ = a , {\displaystyle a_{r}=-v\theta ,\quad a_{\theta }=a,} are called, respectively, the radial and tangential components of acceleration. Velocity Velocity is the speed in combination with the direction of motion of an object . Velocity is a fundamental concept in kinematics ,

1100-439: A x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{(\Delta t)^{2}\to 0}{\frac {\Delta \mathbf {r} }{(\Delta t)^{2}}}={\frac {{\text{d}}^{2}\mathbf {r} }{{\text{d}}t^{2}}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Thus, acceleration

1210-616: A y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Alternatively, a = lim ( Δ t ) 2 → 0 Δ r ( Δ t ) 2 = d 2 r d t 2 =

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1320-638: A ¯ x x ^ + a ¯ y y ^ + a ¯ z z ^ {\displaystyle \mathbf {\bar {a}} ={\frac {\Delta \mathbf {\bar {v}} }{\Delta t}}={\frac {\Delta {\bar {v}}_{x}}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta {\bar {v}}_{y}}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta {\bar {v}}_{z}}{\Delta t}}{\hat {\mathbf {z} }}={\bar {a}}_{x}{\hat {\mathbf {x} }}+{\bar {a}}_{y}{\hat {\mathbf {y} }}+{\bar {a}}_{z}{\hat {\mathbf {z} }}\,} where Δ v

1430-564: A + v ω ) θ ^ + a z z ^ . {\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\theta ){\hat {\mathbf {r} }}+(a+v\omega ){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.} The term − v θ r ^ {\displaystyle -v\theta {\hat {\mathbf {r} }}} acts toward

1540-486: A t t = 1 2 a t 2 = a t 2 2 {\textstyle A={\frac {1}{2}}BH={\frac {1}{2}}att={\frac {1}{2}}at^{2}={\frac {at^{2}}{2}}} . Adding v 0 t {\displaystyle v_{0}t} and a t 2 2 {\textstyle {\frac {at^{2}}{2}}} results in the equation Δ r {\displaystyle \Delta r} results in

1650-452: A robotic arm or the human skeleton . Geometric transformations, also called rigid transformations , are used to describe the movement of components in a mechanical system , simplifying the derivation of the equations of motion. They are also central to dynamic analysis . Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find

1760-524: A very short period of time, is called instantaneous speed . By looking at a speedometer , one can read the instantaneous speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance (25 km). If it continued for only one minute, it would cover about 833 m. In mathematical terms,

1870-413: A 4-hour trip, the distance covered is found to be 320 kilometres. Expressed in graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval covered by the chord. Average speed of an object is Vav = s÷t Speed denotes only how fast an object

1980-417: A constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration. Since the derivative of the position with respect to time gives the change in position (in metres ) divided by the change in time (in seconds ), velocity is measured in metres per second (m/s). Velocity

2090-628: A convenient form. Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F . As the particle moves, its coordinate vector r ( t ) traces its trajectory, which is a curve in space, given by: r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x̂ , ŷ , and ẑ are

2200-954: A particle trajectory on a circular cylinder occurs when there is no movement along the z axis: r ( t ) = r r ^ + z z ^ , {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }},} where r and z 0 are constants. In this case, the velocity v P is given by: v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ = v θ ^ , {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}=v{\hat {\mathbf {\theta } }},} where ω {\displaystyle \omega }

2310-632: A particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The speed of an object is the magnitude of its velocity. It is a scalar quantity: v = | v | = d s d t , {\displaystyle v=|\mathbf {v} |={\frac {{\text{d}}s}{{\text{d}}t}},} where s {\displaystyle s}

SECTION 20

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2420-819: A reference frame. The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector r {\displaystyle {\bf {r}}} can be expressed as r = ( x , y , z ) = x x ^ + y y ^ + z z ^ , {\displaystyle \mathbf {r} =(x,y,z)=x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }},} where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are

2530-417: A straight line), this can be simplified to v = s / t {\displaystyle v=s/t} . The average speed over a finite time interval is the total distance travelled divided by the time duration. Different from instantaneous speed, average speed is defined as the total distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour,

2640-551: A time interval is defined as the ratio. a ¯ = Δ v ¯ Δ t = Δ v ¯ x Δ t x ^ + Δ v ¯ y Δ t y ^ + Δ v ¯ z Δ t z ^ =

2750-588: A two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as v x = d x / d t , {\displaystyle v_{x}=dx/dt,} v y = d y / d t . {\displaystyle v_{y}=dy/dt.} The two-dimensional velocity vector is then defined as v =< v x , v y > {\displaystyle {\textbf {v}}=<v_{x},v_{y}>} . The magnitude of this vector represents speed and

2860-402: A velocity vector, denotes only how fast an object is moving, while velocity indicates both an object's speed and direction. To have a constant velocity , an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at

2970-468: Is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force, F D {\displaystyle F_{D}} , is dependent on the square of velocity and is given as F D = 1 2 ρ v 2 C D A {\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A} where Escape velocity

3080-676: Is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an acceleration . The average velocity of an object over a period of time is its change in position , Δ s {\displaystyle \Delta s} , divided by the duration of the period, Δ t {\displaystyle \Delta t} , given mathematically as v ¯ = Δ s Δ t . {\displaystyle {\bar {v}}={\frac {\Delta s}{\Delta t}}.} The instantaneous velocity of an object

3190-417: Is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time ( x vs. t ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point , and

3300-720: Is defined as v =< v x , v y , v z > {\displaystyle {\textbf {v}}=<v_{x},v_{y},v_{z}>} with its magnitude also representing speed and being determined by | v | = v x 2 + v y 2 + v z 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.} While some textbooks use subscript notation to define Cartesian components of velocity, others use u {\displaystyle u} , v {\displaystyle v} , and w {\displaystyle w} for

3410-401: Is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the x -axis and north is in the direction of the y -axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If

Speed - Misplaced Pages Continue

3520-501: Is defined as the rate of change of position with respect to time, which may also be referred to as the instantaneous velocity to emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v ( t ) , over some time period Δ t . Average velocity can be calculated as: The average velocity

3630-435: Is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration . As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v ( t ) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time:

3740-483: Is found by the distance formula as | v | = v x 2 + v y 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}}}.} In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as v z = d z / d t . {\displaystyle v_{z}=dz/dt.} The three-dimensional velocity vector

3850-635: Is given by the harmonic mean of the speeds v ¯ = n ( 1 v 1 + 1 v 2 + 1 v 3 + ⋯ + 1 v n ) − 1 = n ( ∑ i = 1 n 1 v i ) − 1 . {\displaystyle {\bar {v}}=n\left({1 \over v_{1}}+{1 \over v_{2}}+{1 \over v_{3}}+\dots +{1 \over v_{n}}\right)^{-1}=n\left(\sum _{i=1}^{n}{\frac {1}{v_{i}}}\right)^{-1}.} Although velocity

3960-461: Is judged to be more rapid than another when at a given moment the first object is behind and a moment or so later ahead of the other object." Kinematics Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering , robotics , and biomechanics , kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine ,

4070-459: Is moving, whereas velocity describes both how fast and in which direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be discerned when considering movement around a circle . When something moves in a circular path and returns to its starting point, its average velocity

4180-474: Is position and r ^ {\displaystyle {\hat {\boldsymbol {r}}}} is the radial direction. The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular speed ω {\displaystyle \omega } and

4290-602: Is simply the difference between their positions which is the difference between the components of their position vectors. If point A has position components r A = ( x A , y A , z A ) {\displaystyle \mathbf {r} _{A}=\left(x_{A},y_{A},z_{A}\right)} and point B has position components r B = ( x B , y B , z B ) {\displaystyle \mathbf {r} _{B}=\left(x_{B},y_{B},z_{B}\right)} then

4400-882: Is simply the difference between their velocities v A / B = v A − v B {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}} which is the difference between the components of their velocities. If point A has velocity components v A = ( v A x , v A y , v A z ) {\displaystyle \mathbf {v} _{A}=\left(v_{A_{x}},v_{A_{y}},v_{A_{z}}\right)} and point B has velocity components v B = ( v B x , v B y , v B z ) {\displaystyle \mathbf {v} _{B}=\left(v_{B_{x}},v_{B_{y}},v_{B_{z}}\right)} then

4510-436: Is speed, d {\displaystyle d} is distance, and t {\displaystyle t} is time. A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion often have variations in speed (a car might travel along a street at 50 km/h, slow to 0 km/h, and then reach 30 km/h). Speed at some instant, or assumed constant during

Speed - Misplaced Pages Continue

4620-594: Is the angular velocity of the unit vector θ around the z axis of the cylinder. The acceleration a P of the particle P is now given by: a P = d ( v θ ^ ) d t = a θ ^ − v θ r ^ . {\displaystyle \mathbf {a} _{P}={\frac {{\text{d}}(v{\hat {\mathbf {\theta } }})}{{\text{d}}t}}=a{\hat {\mathbf {\theta } }}-v\theta {\hat {\mathbf {r} }}.} The components

4730-409: Is the gravitational constant and g is the gravitational acceleration . The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of

4840-402: Is the limit of the average speed as the duration of the time interval approaches zero. Speed is the magnitude of velocity (a vector), which indicates additionally the direction of motion. Speed has the dimensions of distance divided by time. The SI unit of speed is the metre per second (m/s), but the most common unit of speed in everyday usage is the kilometre per hour (km/h) or, in

4950-417: Is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, d s / d t {\displaystyle {\text{d}}s/{\text{d}}t} is non-negative, which implies that speed is also non-negative. The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both

5060-411: Is the area under a velocity–time graph. We can take Δ r {\displaystyle \Delta r} by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the A ⋅ B {\displaystyle A\cdot B} where A {\displaystyle A} is the width and B {\displaystyle B}

5170-441: Is the average velocity and Δ t is the time interval. The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative, a = lim Δ t → 0 Δ v Δ t = d v d t = a x x ^ +

5280-769: Is the component of velocity along a circle centered at the origin. v = v T + v R {\displaystyle {\boldsymbol {v}}={\boldsymbol {v}}_{T}+{\boldsymbol {v}}_{R}} where The radial speed (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction. v R = v ⋅ r | r | = v ⋅ r ^ {\displaystyle v_{R}={\frac {{\boldsymbol {v}}\cdot {\boldsymbol {r}}}{\left|{\boldsymbol {r}}\right|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {r}}}} where r {\displaystyle {\boldsymbol {r}}}

5390-948: Is the displacement vector during the time interval Δ t {\displaystyle \Delta t} . In the limit that the time interval Δ t {\displaystyle \Delta t} approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ t → 0 Δ r Δ t = d r d t = v x x ^ + v y y ^ + v z z ^ . {\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {{\text{d}}\mathbf {r} }{{\text{d}}t}}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}.} Thus,

5500-406: Is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The magnitude of the acceleration of an object is the magnitude | a | of its acceleration vector. It is a scalar quantity: |

5610-423: Is the height. In this case A = t {\displaystyle A=t} and B = v 0 {\displaystyle B=v_{0}} (the A {\displaystyle A} here is different from the acceleration a {\displaystyle a} ). This means that the bottom area is t v 0 {\displaystyle tv_{0}} . Now let's find

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5720-399: Is the length of the path (also known as the distance) travelled until time t {\displaystyle t} , the speed equals the time derivative of s {\displaystyle s} : v = d s d t . {\displaystyle v={\frac {ds}{dt}}.} In the special case where the velocity is constant (that is, constant speed in

5830-546: Is the limit average velocity as the time interval approaches zero. At any particular time t , it can be calculated as the derivative of the position with respect to time: v = lim Δ t → 0 Δ s Δ t = d s d t . {\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.} From this derivative equation, in

5940-418: Is the mass of the object. The kinetic energy of a moving object is dependent on its velocity and is given by the equation E k = 1 2 m v 2 {\displaystyle E_{\text{k}}={\tfrac {1}{2}}mv^{2}} where E k is the kinetic energy. Kinetic energy is a scalar quantity as it depends on the square of the velocity. In fluid dynamics , drag

6050-461: Is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. L = m r v T = m r 2 ω {\displaystyle L=mrv_{T}=mr^{2}\omega } where The expression m r 2 {\displaystyle mr^{2}}

6160-547: Is the minimum speed a ballistic object needs to escape from a massive body such as Earth. It represents the kinetic energy that, when added to the object's gravitational potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is v e = 2 G M r = 2 g r , {\displaystyle v_{\text{e}}={\sqrt {\frac {2GM}{r}}}={\sqrt {2gr}},} where G

6270-463: Is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). The radial and traverse velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity

6380-453: Is the speed of light. Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. Consider an object A moving with velocity vector v and an object B with velocity vector w ; these absolute velocities are typically expressed in

6490-420: Is zero, but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by considering only the displacement between the starting and end points, whereas the average speed considers only the total distance travelled. Units of speed include: (* = approximate values) According to Jean Piaget ,

6600-436: The x {\displaystyle x} -, y {\displaystyle y} -, and z {\displaystyle z} -axes respectively. In polar coordinates , a two-dimensional velocity is described by a radial velocity , defined as the component of velocity away from or toward the origin, and a transverse velocity , perpendicular to the radial one. Both arise from angular velocity , which

6710-602: The Cartesian coordinates and x ^ {\displaystyle {\hat {\mathbf {x} }}} , y ^ {\displaystyle {\hat {\mathbf {y} }}} and z ^ {\displaystyle {\hat {\mathbf {z} }}} are the unit vectors along the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} coordinate axes, respectively. The magnitude of

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6820-473: The dot product , which is appropriate as the products are scalars rather than vectors. 2 ( r − r 0 ) ⋅ a = | v | 2 − | v 0 | 2 . {\displaystyle 2\left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} The dot product can be replaced by

6930-842: The unit vectors along the x , y and z axes of the reference frame F , respectively. Consider a particle P that moves only on the surface of a circular cylinder r ( t ) = constant, it is possible to align the z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the x – y plane can be used to define the trajectory as, r ( t ) = r cos ⁡ ( θ ( t ) ) x ^ + r sin ⁡ ( θ ( t ) ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where

7040-504: The French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek γρᾰ́φω grapho ("to write"). Particle kinematics is the study of the trajectory of particles. The position of a particle

7150-572: The US and the UK, miles per hour (mph). For air and marine travel, the knot is commonly used. The fastest possible speed at which energy or information can travel, according to special relativity , is the speed of light in vacuum c = 299 792 458 metres per second (approximately 1 079 000 000  km/h or 671 000 000  mph ). Matter cannot quite reach the speed of light, as this would require an infinite amount of energy. In relativity physics,

7260-424: The acceleration a P , which is the time derivative of the velocity v P , is given by: a P = d d t ( v r ^ + v θ ^ + v z z ^ ) = ( a − v θ ) r ^ + (

7370-729: The acceleration is constant, a = Δ v Δ t = v − v 0 t {\displaystyle \mathbf {a} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} -\mathbf {v} _{0}}{t}}} can be substituted into the above equation to give: r ( t ) = r 0 + ( v + v 0 2 ) t . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\left({\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\right)t.} A relationship between velocity, position and acceleration without explicit time dependence can be had by solving

7480-739: The average acceleration for time and substituting and simplifying t = v − v 0 a {\displaystyle t={\frac {\mathbf {v} -\mathbf {v} _{0}}{\mathbf {a} }}} ( r − r 0 ) ⋅ a = ( v − v 0 ) ⋅ v + v 0 2   , {\displaystyle \left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =\left(\mathbf {v} -\mathbf {v} _{0}\right)\cdot {\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\ ,} where ⋅ {\displaystyle \cdot } denotes

7590-423: The average speed is 80 kilometres per hour. Likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres (km) is divided by a time in hours (h), the result is in kilometres per hour (km/h). Average speed does not describe the speed variations that may have taken place during shorter time intervals (as it is the entire distance covered divided by

7700-1574: The average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity. If t 1 = t 2 = t 3 = ... = t , then average speed is given by the arithmetic mean of the speeds v ¯ = v 1 + v 2 + v 3 + ⋯ + v n n = 1 n ∑ i = 1 n v i {\displaystyle {\bar {v}}={v_{1}+v_{2}+v_{3}+\dots +v_{n} \over n}={\frac {1}{n}}\sum _{i=1}^{n}{v_{i}}} v ¯ = s 1 + s 2 + s 3 + ⋯ + s n t 1 + t 2 + t 3 + ⋯ + t n = s 1 + s 2 + s 3 + ⋯ + s n s 1 v 1 + s 2 v 2 + s 3 v 3 + ⋯ + s n v n {\displaystyle {\bar {v}}={s_{1}+s_{2}+s_{3}+\dots +s_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}={{s_{1}+s_{2}+s_{3}+\dots +s_{n}} \over {{s_{1} \over v_{1}}+{s_{2} \over v_{2}}+{s_{3} \over v_{3}}+\dots +{s_{n} \over v_{n}}}}} If s 1 = s 2 = s 3 = ... = s , then average speed

7810-456: The base body as long as it does not intersect with something in its path. In special relativity , the dimensionless Lorentz factor appears frequently, and is given by γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} where γ is the Lorentz factor and c

7920-456: The branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity : both magnitude and direction are needed to define it. The scalar absolute value ( magnitude ) of velocity is called speed , being a coherent derived unit whose quantity is measured in the SI ( metric system ) as metres per second (m/s or m⋅s ). For example, "5 metres per second"

8030-635: The case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors ( α ) is 0, so cos ⁡ 0 = 1 {\displaystyle \cos 0=1} , and | v | 2 = | v 0 | 2 + 2 | a | | r − r 0 | . {\displaystyle |\mathbf {v} |^{2}=|\mathbf {v} _{0}|^{2}+2\left|\mathbf {a} \right|\left|\mathbf {r} -\mathbf {r} _{0}\right|.} This can be simplified using

8140-460: The center of curvature of the path at that point on the path, is commonly called the centripetal acceleration . The term v ω θ ^ {\displaystyle v\omega {\hat {\mathbf {\theta } }}} is called the Coriolis acceleration . If the trajectory of the particle is constrained to lie on a cylinder, then the radius r is constant and

8250-412: The components of their accelerations. If point C has acceleration components a C = ( a C x , a C y , a C z ) {\displaystyle \mathbf {a} _{C}=\left(a_{C_{x}},a_{C_{y}},a_{C_{z}}\right)} and point B has acceleration components a B = (

8360-444: The concept of rapidity replaces the classical idea of speed. Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, that is v = d t , {\displaystyle v={\frac {d}{t}},} where v {\displaystyle v}

8470-402: The concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. While the terms speed and velocity are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, the scalar magnitude of

8580-2770: The constant distance from the center is denoted as r , and θ ( t ) is a function of time. The cylindrical coordinates for r ( t ) can be simplified by introducing the radial and tangential unit vectors, r ^ = cos ⁡ ( θ ( t ) ) x ^ + sin ⁡ ( θ ( t ) ) y ^ , θ ^ = − sin ⁡ ( θ ( t ) ) x ^ + cos ⁡ ( θ ( t ) ) y ^ . {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.} and their time derivatives from elementary calculus: d r ^ d t = ω θ ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.} d 2 r ^ d t 2 = d ( ω θ ^ ) d t = α θ ^ − ω r ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega {\hat {\mathbf {r} }}.} d θ ^ d t = − θ r ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\theta {\hat {\mathbf {r} }}.} d 2 θ ^ d t 2 = d ( − θ r ^ ) d t = − α r ^ − ω 2 θ ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\theta {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.} Using this notation, r ( t ) takes

8690-590: The constant tangential acceleration is applied along that path , so v 2 = v 0 2 + 2 a Δ r . {\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.} This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that Δ r = ∫ v d t {\textstyle \Delta r=\int v\,{\text{d}}t} or Δ r {\displaystyle \Delta r}

8800-574: The cosine of the angle α between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case: 2 | r − r 0 | | a | cos ⁡ α = | v | 2 − | v 0 | 2 . {\displaystyle 2\left|\mathbf {r} -\mathbf {r} _{0}\right|\left|\mathbf {a} \right|\cos \alpha =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} In

8910-449: The equation Δ r = v 0 t + a t 2 2 {\textstyle \Delta r=v_{0}t+{\frac {at^{2}}{2}}} . This equation is applicable when the final velocity v is unknown. It is often convenient to formulate the trajectory of a particle r ( t ) = ( x ( t ), y ( t ), z ( t )) using polar coordinates in the X – Y plane. In this case, its velocity and acceleration take

9020-780: The form, r ( t ) = r r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} In general, the trajectory r ( t ) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r ( t ) = r ( t ) r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} Where r , θ , and z might be continuously differentiable functions of time and

9130-972: The function notation is dropped for simplicity. The velocity vector v P is the time derivative of the trajectory r ( t ), which yields: v P = d d t ( r r ^ + z z ^ ) = v r ^ + r ω θ ^ + v z z ^ = v ( r ^ + θ ^ ) + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.} Similarly,

9240-540: The inertial frame chosen is that in which the latter of the two mentioned objects is in rest. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame. In the one-dimensional case, the velocities are scalars and the equation is either: v rel = v − ( − w ) , {\displaystyle v_{\text{rel}}=v-(-w),} if

9350-677: The instantaneous speed v {\displaystyle v} is defined as the magnitude of the instantaneous velocity v {\displaystyle {\boldsymbol {v}}} , that is, the derivative of the position r {\displaystyle {\boldsymbol {r}}} with respect to time : v = | v | = | r ˙ | = | d r d t | . {\displaystyle v=\left|{\boldsymbol {v}}\right|=\left|{\dot {\boldsymbol {r}}}\right|=\left|{\frac {d{\boldsymbol {r}}}{dt}}\right|\,.} If s {\displaystyle s}

9460-432: The intuition for the notion of speed in humans precedes that of duration, and is based on the notion of outdistancing. Piaget studied this subject inspired by a question asked to him in 1928 by Albert Einstein : "In what order do children acquire the concepts of time and speed?" Children's early concept of speed is based on "overtaking", taking only temporal and spatial orders into consideration, specifically: "A moving object

9570-438: The notation for the magnitudes of the vectors | a | = a , | v | = v , | r − r 0 | = Δ r {\displaystyle |\mathbf {a} |=a,|\mathbf {v} |=v,|\mathbf {r} -\mathbf {r} _{0}|=\Delta r} where Δ r {\displaystyle \Delta r} can be any curvaceous path taken as

9680-477: The one-dimensional case it can be seen that the area under a velocity vs. time ( v vs. t graph) is the displacement, s . In calculus terms, the integral of the velocity function v ( t ) is the displacement function s ( t ) . In the figure, this corresponds to the yellow area under the curve. s = ∫ v   d t . {\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.} Although

9790-410: The particle's position as a function of time. The velocity of a particle is a vector quantity that describes the direction as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle ( displacement ) by

9900-558: The position of point A relative to point B is the difference between their components: r A / B = r A − r B = ( x A − x B , y A − y B , z A − z B ) {\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}=\left(x_{A}-x_{B},y_{A}-y_{B},z_{A}-z_{B}\right)} The velocity of one point relative to another

10010-439: The position vector | r | {\displaystyle \left|\mathbf {r} \right|} gives the distance between the point r {\displaystyle \mathbf {r} } and the origin. | r | = x 2 + y 2 + z 2 . {\displaystyle |\mathbf {r} |={\sqrt {x^{2}+y^{2}+z^{2}}}.} The direction cosines of

10120-993: The position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector. The trajectory of a particle is a vector function of time, r ( t ) {\displaystyle \mathbf {r} (t)} , which defines the curve traced by the moving particle, given by r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x ( t ) {\displaystyle x(t)} , y ( t ) {\displaystyle y(t)} , and z ( t ) {\displaystyle z(t)} describe each coordinate of

10230-754: The radius (the magnitude of the position). v T = | r × v | | r | = v ⋅ t ^ = ω | r | {\displaystyle v_{T}={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {t}}}=\omega |{\boldsymbol {r}}|} such that ω = | r × v | | r | 2 . {\displaystyle \omega ={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|^{2}}}.} Angular momentum in scalar form

10340-649: The range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A.M. Ampère 's cinématique , which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move"). Kinematic and cinématique are related to

10450-399: The rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over

10560-710: The same inertial reference frame . Then, the velocity of object A relative to object B is defined as the difference of the two velocity vectors: v A  relative to  B = v − w {\displaystyle {\boldsymbol {v}}_{A{\text{ relative to }}B}={\boldsymbol {v}}-{\boldsymbol {w}}} Similarly, the relative velocity of object B moving with velocity w , relative to object A moving with velocity v is: v B  relative to  A = w − v {\displaystyle {\boldsymbol {v}}_{B{\text{ relative to }}A}={\boldsymbol {w}}-{\boldsymbol {v}}} Usually,

10670-403: The second time derivative of the relative position vector r B/A . Assuming that the initial conditions of the position, r 0 {\displaystyle \mathbf {r} _{0}} , and velocity v 0 {\displaystyle \mathbf {v} _{0}} at time t = 0 {\displaystyle t=0} are known, the first integration yields

10780-432: The special case of constant acceleration, velocity can be studied using the suvat equations . By considering a as being equal to some arbitrary constant vector, this shows v = u + a t {\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t} with v as the velocity at time t and u as the velocity at time t = 0 . By combining this equation with

10890-415: The suvat equation x = u t + a t /2 , it is possible to relate the displacement and the average velocity by x = ( u + v ) 2 t = v ¯ t . {\displaystyle {\boldsymbol {x}}={\frac {({\boldsymbol {u}}+{\boldsymbol {v}})}{2}}t={\boldsymbol {\bar {v}}}t.} It is also possible to derive an expression for

11000-404: The time derivative of the relative position vector r B/A . The acceleration of one point C relative to another point B is simply the difference between their accelerations. a C / B = a C − a B {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}} which is the difference between

11110-1152: The time interval. This ratio is called the average velocity over that time interval and is defined as v ¯ = Δ r Δ t = Δ x Δ t x ^ + Δ y Δ t y ^ + Δ z Δ t z ^ = v ¯ x x ^ + v ¯ y y ^ + v ¯ z z ^ {\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta z}{\Delta t}}{\hat {\mathbf {z} }}={\bar {v}}_{x}{\hat {\mathbf {x} }}+{\bar {v}}_{y}{\hat {\mathbf {y} }}+{\bar {v}}_{z}{\hat {\mathbf {z} }}\,} where Δ r {\displaystyle \Delta \mathbf {r} }

11220-471: The top area (a triangle). The area of a triangle is 1 2 B H {\textstyle {\frac {1}{2}}BH} where B {\displaystyle B} is the base and H {\displaystyle H} is the height. In this case, B = t {\displaystyle B=t} and H = a t {\displaystyle H=at} or A = 1 2 B H = 1 2

11330-431: The total time of travel), and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to d = v ¯ t . {\displaystyle d={\boldsymbol {\bar {v}}}t\,.} Using this equation for an average speed of 80 kilometres per hour on

11440-481: The tower is 50 m high, and this height is measured along the z -axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m). In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to

11550-401: The two objects are moving in opposite directions, or: v rel = v − ( + w ) , {\displaystyle v_{\text{rel}}=v-(+w),} if the two objects are moving in the same direction. In multi-dimensional Cartesian coordinate systems , velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In

11660-555: The value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated. In classical mechanics, Newton's second law defines momentum , p, as a vector that is the product of an object's mass and velocity, given mathematically as p = m v {\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}} where m

11770-803: The velocity and acceleration vectors simplify. The velocity of v P is the time derivative of the trajectory r ( t ), v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ + v z z ^ = v θ ^ + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}.} A special case of

11880-654: The velocity independent of time, known as the Torricelli equation , as follows: v 2 = v ⋅ v = ( u + a t ) ⋅ ( u + a t ) = u 2 + 2 t ( a ⋅ u ) + a 2 t 2 {\displaystyle v^{2}={\boldsymbol {v}}\cdot {\boldsymbol {v}}=({\boldsymbol {u}}+{\boldsymbol {a}}t)\cdot ({\boldsymbol {u}}+{\boldsymbol {a}}t)=u^{2}+2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}} ( 2

11990-676: The velocity of point A relative to point B is the difference between their components: v A / B = v A − v B = ( v A x − v B x , v A y − v B y , v A z − v B z ) {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}=\left(v_{A_{x}}-v_{B_{x}},v_{A_{y}}-v_{B_{y}},v_{A_{z}}-v_{B_{z}}\right)} Alternatively, this same result could be obtained by computing

12100-683: The velocity of the particle as a function of time. v ( t ) = v 0 + ∫ 0 t a d τ = v 0 + a t . {\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{0}^{t}\mathbf {a} \,{\text{d}}\tau =\mathbf {v} _{0}+\mathbf {a} t.} A second integration yields its path (trajectory), r ( t ) = r 0 + ∫ 0 t v ( τ ) d τ = r 0 + ∫ 0 t ( v 0 +

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