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61-519: The mass might be a projectile or a satellite . For example, it can be an orbit — the path of a planet , asteroid , or comet as it travels around a central mass . In control theory , a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map ). In discrete mathematics , a trajectory is a sequence ( f k ( x ) ) k ∈ N {\displaystyle (f^{k}(x))_{k\in \mathbb {N} }} of values calculated by

122-649: A mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x 1 , ..., x n and have masses m 1 , ..., m n , then the potential of the distribution at the point x is V ( x ) = ∑ i = 1 n − G m i ‖ x − x i ‖ . {\displaystyle V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{\|\mathbf {x} -\mathbf {x} _{i}\|}}.} If

183-439: A combination of these mechanisms. Railguns utilize electromagnetic fields to provide a constant acceleration along the entire length of the device, greatly increasing the muzzle velocity . Some projectiles provide propulsion during flight by means of a rocket engine or jet engine . In military terminology, a rocket is unguided, while a missile is guided . Note the two meanings of "rocket" (weapon and engine): an ICBM

244-402: A given initial speed v {\displaystyle v} is obtained when v h = v v {\displaystyle v_{h}=v_{v}} , i.e. the initial angle is 45 ∘ {\displaystyle ^{\circ }} . This range is v 2 / g {\displaystyle v^{2}/g} , and the maximum altitude at

305-678: A given range d h {\displaystyle d_{h}} . The angle θ {\displaystyle \theta } giving the maximum range can be found by considering the derivative or R {\displaystyle R} with respect to θ {\displaystyle \theta } and setting it to zero. which has a nontrivial solution at 2 θ = π / 2 = 90 ∘ {\displaystyle 2\theta =\pi /2=90^{\circ }} , or θ = 45 ∘ {\displaystyle \theta =45^{\circ }} . The maximum range

366-421: A location is the gravitational potential energy ( U ) at that location per unit mass: V = U m , {\displaystyle V={\frac {U}{m}},} where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity. If the body has a mass of 1 kilogram, then

427-499: A player is positioned so as to catch it as it descends, he sees its angle of elevation increasing continuously throughout its flight. The tangent of the angle of elevation is proportional to the time since the ball was sent into the air, usually by being struck with a bat. Even when the ball is really descending, near the end of its flight, its angle of elevation seen by the player continues to increase. The player therefore sees it as if it were ascending vertically at constant speed. Finding

488-440: A point x is given by V ( x ) = − ∫ R 3 G | x − r |   d m ( r ) . {\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {r} |}}\ dm(\mathbf {r} ).} The potential can be expanded in a series of Legendre polynomials . Represent

549-482: A rock that is thrown for short distances, for example at the surface of the Moon . In this simple approximation, the trajectory takes the shape of a parabola . Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance ( drag and aerodynamics ). This is the focus of the discipline of ballistics . One of the remarkable achievements of Newtonian mechanics

610-687: A specific angle θ {\displaystyle \theta } : 1. Time to reach maximum height. It is symbolized as ( t {\displaystyle t} ), which is the time taken for the projectile to reach the maximum height from the plane of projection. Mathematically, it is given as t = U sin ⁡ θ / g {\displaystyle t=U\sin \theta /g} where g {\displaystyle g} = acceleration due to gravity (app 9.81 m/s²), U {\displaystyle U} = initial velocity (m/s) and θ {\displaystyle \theta } = angle made by

671-490: A uniform downwards gravitational force we consider two bodies orbiting with the mutual gravitation between them, we obtain Kepler's laws of planetary motion . The derivation of these was one of the major works of Isaac Newton and provided much of the motivation for the development of differential calculus . If a projectile, such as a baseball or cricket ball, travels in a parabolic path, with negligible air resistance, and if

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732-574: A uniform gravitational field in the absence of other forces (such as air drag) was first investigated by Galileo Galilei . To neglect the action of the atmosphere in shaping a trajectory would have been considered a futile hypothesis by practical-minded investigators all through the Middle Ages in Europe . Nevertheless, by anticipating the existence of the vacuum , later to be demonstrated on Earth by his collaborator Evangelista Torricelli , Galileo

793-587: A unit mass in from infinity to that point: V ( x ) = W m = 1 m ∫ ∞ x F ⋅ d x = 1 m ∫ ∞ x G m M x 2 d x = − G M x , {\displaystyle V(\mathbf {x} )={\frac {W}{m}}={\frac {1}{m}}\int _{\infty }^{x}\mathbf {F} \cdot d\mathbf {x} ={\frac {1}{m}}\int _{\infty }^{x}{\frac {GmM}{x^{2}}}dx=-{\frac {GM}{x}},} where G

854-508: Is a projectile weapon based solely on a projectile's kinetic energy to inflict damage to a target, instead of using any explosive , incendiary / thermal , chemical or radiological payload . All kinetic weapons work by attaining a high flight speed — generally supersonic or even up to hypervelocity — and collide with their targets, converting their kinetic energy and relative impulse into destructive shock waves , heat and cavitation . In kinetic weapons with unpowered flight ,

915-429: Is a guided missile with a rocket engine. An explosion, whether or not by a weapon, causes the debris to act as multiple high velocity projectiles. An explosive weapon or device may also be designed to produce many high velocity projectiles by the break-up of its casing; these are correctly termed fragments . In projectile motion the most important force applied to the ‘projectile’ is the propelling force, in this case

976-463: Is a potential function coming from a continuous mass distribution ρ ( r ), then ρ can be recovered using the Laplace operator , Δ : ρ ( x ) = 1 4 π G Δ V ( x ) . {\displaystyle \rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).} This holds pointwise whenever ρ is continuous and

1037-514: Is a vector of length x pointing from the point mass toward the small body and x ^ {\displaystyle {\hat {\mathbf {x} }}} is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law : ‖ a ‖ = G M x 2 . {\displaystyle \|\mathbf {a} \|={\frac {GM}{x^{2}}}.} The potential associated with

1098-583: Is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance. Their similarity is correlated with both associated fields having conservative forces . Mathematically, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory . It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies. The gravitational potential ( V ) at

1159-737: Is the distance between the points x and r . If there is a function ρ ( r ) representing the density of the distribution at r , so that dm ( r ) = ρ ( r ) dv ( r ) , where dv ( r ) is the Euclidean volume element , then the gravitational potential is the volume integral V ( x ) = − ∫ R 3 G ‖ x − r ‖ ρ ( r ) d v ( r ) . {\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).} If V

1220-464: Is the gravitational constant , and F is the gravitational force. The product GM is the standard gravitational parameter and is often known to higher precision than G or M separately. The potential has units of energy per mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero. The gravitational field , and thus

1281-422: Is the acceleration due to gravity). The range , R , is the greatest distance the object travels along the x-axis in the I sector. The initial velocity , v i , is the speed at which said object is launched from the point of origin. The initial angle , θ i , is the angle at which said object is released. The g is the respective gravitational pull on the object within a null-medium. The height , h ,

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1342-815: Is the component of the center of mass in the x direction; this vanishes because the vector x emanates from the center of mass. So, bringing the integral under the sign of the summation gives V ( x ) = − G M | x | − G | x | ∫ ( r | x | ) 2 3 cos 2 ⁡ θ − 1 2 d m ( r ) + ⋯ {\displaystyle V(\mathbf {x} )=-{\frac {GM}{|\mathbf {x} |}}-{\frac {G}{|\mathbf {x} |}}\int \left({\frac {r}{|\mathbf {x} |}}\right)^{2}{\frac {3\cos ^{2}\theta -1}{2}}dm(\mathbf {r} )+\cdots } This shows that elongation of

1403-464: Is the greatest parabolic height said object reaches within its trajectory In terms of angle of elevation θ {\displaystyle \theta } and initial speed v {\displaystyle v} : giving the range as This equation can be rearranged to find the angle for a required range Note that the sine function is such that there are two solutions for θ {\displaystyle \theta } for

1464-433: Is the maximum height attained by the projectile OR the maximum displacement on the vertical axis (y-axis) covered by the projectile. It is given as H = U 2 sin 2 ⁡ θ / 2 g {\displaystyle H=U^{2}\sin ^{2}\theta /2g} . 4. Range ( R {\displaystyle R} ): The Range of a projectile is the horizontal distance covered (on

1525-403: Is then R max = v 2 / g {\displaystyle R_{\max }=v^{2}/g\,} . At this angle sin ⁡ ( π / 2 ) = 1 {\displaystyle \sin(\pi /2)=1} , so the maximum height obtained is v 2 4 g {\displaystyle {v^{2} \over 4g}} . To find

1586-507: Is zero outside of a bounded set. In general, the mass measure dm can be recovered in the same way if the Laplace operator is taken in the sense of distributions . As a consequence, the gravitational potential satisfies Poisson's equation . See also Green's function for the three-variable Laplace equation and Newtonian potential . The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including

1647-400: Is zero when θ = π / 2 = 90 ∘ {\displaystyle \theta =\pi /2=90^{\circ }} . So the maximum height H m a x = v 2 2 g {\displaystyle H_{\mathrm {max} }={v^{2} \over 2g}} is obtained when the projectile is fired straight up. If instead of

1708-401: The gravitational potential is a scalar potential associating with each point in space the work ( energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the conservative gravitational field . It is analogous to the electric potential with mass playing the role of charge . The reference point, where the potential is zero,

1769-724: The muzzle velocity or launch velocity often determines the effective range and potential damage of the kinetic projectile. Kinetic weapons are the oldest and most common ranged weapons used in human history , with the projectiles varying from blunt projectiles such as rocks and round shots , pointed missiles such as arrows , bolts , darts , and javelins , to modern tapered high-velocity impactors such as bullets , flechettes , and penetrators . Typical kinetic weapons accelerate their projectiles mechanically (by muscle power , mechanical advantage devices , elastic energy or pneumatics ) or chemically (by propellant combustion , as with firearms ), but newer technologies are enabling

1830-1831: The Legendre polynomials of degree n . Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in X = cos  θ . So the potential can be expanded in a series that is convergent for positions x such that r < | x | for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system): V ( x ) = − G | x | ∫ ∑ n = 0 ∞ ( r | x | ) n P n ( cos ⁡ θ ) d m ( r ) = − G | x | ∫ ( 1 + ( r | x | ) cos ⁡ θ + ( r | x | ) 2 3 cos 2 ⁡ θ − 1 2 + ⋯ ) d m ( r ) {\displaystyle {\begin{aligned}V(\mathbf {x} )&=-{\frac {G}{|\mathbf {x} |}}\int \sum _{n=0}^{\infty }\left({\frac {r}{|\mathbf {x} |}}\right)^{n}P_{n}(\cos \theta )\,dm(\mathbf {r} )\\&=-{\frac {G}{|\mathbf {x} |}}\int \left(1+\left({\frac {r}{|\mathbf {x} |}}\right)\cos \theta +\left({\frac {r}{|\mathbf {x} |}}\right)^{2}{\frac {3\cos ^{2}\theta -1}{2}}+\cdots \right)\,dm(\mathbf {r} )\end{aligned}}} The integral ∫ r cos ⁡ ( θ ) d m {\textstyle \int r\cos(\theta )\,dm}

1891-499: The Sun, then it is also influenced by other forces such as the solar wind and radiation pressure , which modify the orbit and cause the comet to eject material into space. Newton's theory later developed into the branch of theoretical physics known as classical mechanics . It employs the mathematics of differential calculus (which was also initiated by Newton in his youth). Over the centuries, countless scientists have contributed to

Trajectory - Misplaced Pages Continue

1952-406: The acceleration is a little larger at the poles than at the equator because Earth is an oblate spheroid . Within a spherically symmetric mass distribution, it is possible to solve Poisson's equation in spherical coordinates . Within a uniform spherical body of radius R , density ρ, and mass m , the gravitational force g inside the sphere varies linearly with distance r from the center, giving

2013-600: The acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is a = − G M x 3 x = − G M x 2 x ^ , {\displaystyle \mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},} where x

2074-687: The angle giving the maximum height for a given speed calculate the derivative of the maximum height H = v 2 sin 2 ⁡ ( θ ) / ( 2 g ) {\displaystyle H=v^{2}\sin ^{2}(\theta )/(2g)} with respect to θ {\displaystyle \theta } , that is d H d θ = v 2 2 cos ⁡ ( θ ) sin ⁡ ( θ ) / ( 2 g ) {\displaystyle {\mathrm {d} H \over \mathrm {d} \theta }=v^{2}2\cos(\theta )\sin(\theta )/(2g)} which

2135-408: The body causes a lower potential in the direction of elongation, and a higher potential in perpendicular directions, compared to the potential due to a spherical mass, if we compare cases with the same distance to the center of mass. (If we compare cases with the same distance to the surface , the opposite is true.) The absolute value of gravitational potential at a number of locations with regards to

2196-506: The constant G , with 𝜌 being a constant charge density) to electromagnetism. A spherically symmetric mass distribution behaves to an observer completely outside the distribution as though all of the mass was concentrated at the center, and thus effectively as a point mass , by the shell theorem . On the surface of the earth, the acceleration is given by so-called standard gravity g , approximately 9.8 m/s , although this value varies slightly with latitude and altitude. The magnitude of

2257-408: The development of potential weapons using electromagnetically launched projectiles, such as railguns , coilguns and mass drivers . There are also concept weapons that are accelerated by gravity , as in the case of kinetic bombardment weapons designed for space warfare . Some projectiles stay connected by a cable to the launch equipment after launching it: An object projected at an angle to

2318-493: The development of these two disciplines. Classical mechanics became a most prominent demonstration of the power of rational thought, i.e. reason , in science as well as technology. It helps to understand and predict an enormous range of phenomena ; trajectories are but one example. Consider a particle of mass m {\displaystyle m} , moving in a potential field V {\displaystyle V} . Physically speaking, mass represents inertia , and

2379-435: The difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height: Δ U ≈ m g Δ h . {\displaystyle \Delta U\approx mg\Delta h.} The gravitational potential V at a distance x from a point mass of mass M can be defined as the work W that needs to be done by an external agent to bring

2440-403: The field V {\displaystyle V} represents external forces of a particular kind known as "conservative". Given V {\displaystyle V} at every relevant position, there is a way to infer the associated force that would act at that position, say from gravity. Not all forces can be expressed in this way, however. The motion of the particle is described by

2501-552: The gravitational potential inside the sphere, which is V ( r ) = 2 3 π G ρ [ r 2 − 3 R 2 ] = G m 2 R 3 [ r 2 − 3 R 2 ] , r ≤ R , {\displaystyle V(r)={\frac {2}{3}}\pi G\rho \left[r^{2}-3R^{2}\right]={\frac {Gm}{2R^{3}}}\left[r^{2}-3R^{2}\right],\qquad r\leq R,} which differentiably connects to

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2562-481: The horizontal has both the vertical and horizontal components of velocity. The vertical component of the velocity on the y-axis is given as V y = U sin ⁡ θ {\displaystyle V_{y}=U\sin \theta } while the horizontal component of the velocity is V x = U cos ⁡ θ {\displaystyle V_{x}=U\cos \theta } . There are various calculations for projectiles at

2623-425: The inertial frame the co-ordinates of the projectile becomes y = x tan ⁡ ( θ ) − g ( x / v h ) 2 / 2 {\displaystyle y=x\tan(\theta )-g(x/v_{h})^{2}/2} That is: (where v 0 is the initial velocity, θ {\displaystyle \theta } is the angle of elevation, and g

2684-644: The influence of gravity and air resistance . Although any objects in motion through space are projectiles, they are commonly found in warfare and sports (for example, a thrown baseball , kicked football , fired bullet , shot arrow , stone released from catapult ). In ballistics mathematical equations of motion are used to analyze projectile trajectories through launch, flight , and impact . Blowguns and pneumatic rifles use compressed gases, while most other guns and cannons utilize expanding gases liberated by sudden chemical reactions by propellants like smokeless powder . Light-gas guns use

2745-461: The initial vertical speed be v v = v sin ⁡ ( θ ) {\displaystyle v_{v}=v\sin(\theta )} . It will also be shown that the range is 2 v h v v / g {\displaystyle 2v_{h}v_{v}/g} , and the maximum altitude is v v 2 / 2 g {\displaystyle v_{v}^{2}/2g} . The maximum range for

2806-413: The iterated application of a mapping f {\displaystyle f} to an element x {\displaystyle x} of its source. A familiar example of a trajectory is the path of a projectile, such as a thrown ball or rock. In a significantly simplified model, the object moves only under the influence of a uniform gravitational force field . This can be a good approximation for

2867-872: The last integral, r = | r | and θ is the angle between x and r . (See "mathematical form".) The integrand can be expanded as a Taylor series in Z = r /| x | , by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalized binomial theorem . The resulting series is the generating function for the Legendre polynomials: ( 1 − 2 X Z + Z 2 ) − 1 2   = ∑ n = 0 ∞ Z n P n ( X ) {\displaystyle \left(1-2XZ+Z^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }Z^{n}P_{n}(X)} valid for | X | ≤ 1 and | Z | < 1 . The coefficients P n are

2928-587: The mass distribution is given as a mass measure dm on three-dimensional Euclidean space R , then the potential is the convolution of − G /| r | with dm . In good cases this equals the integral V ( x ) = − ∫ R 3 G ‖ x − r ‖ d m ( r ) , {\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,dm(\mathbf {r} ),} where | x − r |

2989-879: The maximum range is v 2 / ( 4 g ) {\displaystyle v^{2}/(4g)} . Assume the motion of the projectile is being measured from a free fall frame which happens to be at ( x , y ) = (0,0) at  t  = 0. The equation of motion of the projectile in this frame (by the equivalence principle ) would be y = x tan ⁡ ( θ ) {\displaystyle y=x\tan(\theta )} . The co-ordinates of this free-fall frame, with respect to our inertial frame would be y = − g t 2 / 2 {\displaystyle y=-gt^{2}/2} . That is, y = − g ( x / v h ) 2 / 2 {\displaystyle y=-g(x/v_{h})^{2}/2} . Now translating back to

3050-455: The place from which the ball appears to rise steadily helps the player to position himself correctly to make the catch. If he is too close to the batsman who has hit the ball, it will appear to rise at an accelerating rate. If he is too far from the batsman, it will appear to slow rapidly, and then to descend. Projectile A projectile is an object that is propelled by the application of an external force and then moves freely under

3111-532: The point of launch of the projectile. The x {\displaystyle x} -axis is tangent to the ground, and the y {\displaystyle y} axis is perpendicular to it ( parallel to the gravitational field lines ). Let g {\displaystyle g} be the acceleration of gravity . Relative to the flat terrain, let the initial horizontal speed be v h = v cos ⁡ ( θ ) {\displaystyle v_{h}=v\cos(\theta )} and

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3172-1215: The points x and r as position vectors relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give V ( x ) = − ∫ R 3 G | x | 2 − 2 x ⋅ r + | r | 2 d m ( r ) = − 1 | x | ∫ R 3 G 1 − 2 r | x | cos ⁡ θ + ( r | x | ) 2 d m ( r ) {\displaystyle {\begin{aligned}V(\mathbf {x} )&=-\int _{\mathbb {R} ^{3}}{\frac {G}{\sqrt {|\mathbf {x} |^{2}-2\mathbf {x} \cdot \mathbf {r} +|\mathbf {r} |^{2}}}}\,dm(\mathbf {r} )\\&=-{\frac {1}{|\mathbf {x} |}}\int _{\mathbb {R} ^{3}}{\frac {G}{\sqrt {1-2{\frac {r}{|\mathbf {x} |}}\cos \theta +\left({\frac {r}{|\mathbf {x} |}}\right)^{2}}}}\,dm(\mathbf {r} )\end{aligned}}} where, in

3233-530: The potential energy to be assigned to that body is equal to the gravitational potential. So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity. In some situations, the equations can be simplified by assuming a field that is nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration , g , can be considered constant. In that case,

3294-421: The potential function for the outside of the sphere (see the figure at the top). In general relativity , the gravitational potential is replaced by the metric tensor . When the gravitational field is weak and the sources are moving very slowly compared to light-speed, general relativity reduces to Newtonian gravity, and the metric tensor can be expanded in terms of the gravitational potential. The potential at

3355-447: The projectile with the horizontal axis. 2. Time of flight ( T {\displaystyle T} ): this is the total time taken for the projectile to fall back to the same plane from which it was projected. Mathematically it is given as T = 2 U sin ⁡ θ / g {\displaystyle T=2U\sin \theta /g} . 3. Maximum Height ( H {\displaystyle H} ): this

3416-648: The propelling forces are the muscles that act upon the ball to make it move, and the stronger the force applied, the more propelling force, which means the projectile (the ball) will travel farther. See pitching , bowling . Many projectiles, e.g. shells , may carry an explosive charge or another chemical or biological substance. Aside from explosive payload, a projectile can be designed to cause special damage, e.g. fire (see also early thermal weapons ), or poisoning (see also arrow poison ). A kinetic energy weapon (also known as kinetic weapon, kinetic energy warhead, kinetic warhead, kinetic projectile, kinetic kill vehicle)

3477-409: The second-order differential equation On the right-hand side, the force is given in terms of ∇ V {\displaystyle \nabla V} , the gradient of the potential, taken at positions along the trajectory. This is the mathematical form of Newton's second law of motion : force equals mass times acceleration, for such situations. The ideal case of motion of a projectile in

3538-447: The symmetrical and degenerate ones. These include the sphere, where the three semi axes are equal; the oblate (see reference ellipsoid ) and prolate spheroids, where two semi axes are equal; the degenerate ones where one semi axes is infinite (the elliptical and circular cylinder) and the unbounded sheet where two semi axes are infinite. All these shapes are widely used in the applications of the gravitational potential integral (apart from

3599-458: The x-axis) by the projectile. Mathematically, R = U 2 sin ⁡ 2 θ / g {\displaystyle R=U^{2}\sin 2\theta /g} . The Range is maximum when angle θ {\displaystyle \theta } = 45°, i.e. sin ⁡ 2 θ = 1 {\displaystyle \sin 2\theta =1} . Gravitational potential In classical mechanics ,

3660-414: Was able to initiate the future science of mechanics . In a near vacuum, as it turns out for instance on the Moon , his simplified parabolic trajectory proves essentially correct. In the analysis that follows, we derive the equation of motion of a projectile as measured from an inertial frame at rest with respect to the ground. Associated with the frame is a right-hand coordinate system with its origin at

3721-467: Was the derivation of Kepler's laws of planetary motion . In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the Sun ), the trajectory of a moving object is a conic section , usually an ellipse or a hyperbola . This agrees with the observed orbits of planets , comets , and artificial spacecraft to a reasonably good approximation, although if a comet passes close to

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