A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics , a trajectory is defined by Hamiltonian mechanics via canonical coordinates ; hence, a complete trajectory is defined by position and momentum , simultaneously.
32-496: Dowker is a surname. Notable people with the surname include: Clifford Hugh Dowker (1912–1982), Canadian mathematician Fay Dowker (born 1965), British physicist Felicity Dowker (born 1980), Australian fantasy writer Hasted Dowker (1900–1986), Canadian Anglican priest Ray Dowker (1919–2004), New Zealand cricketer Yael Dowker (1919–2016), Israeli-English mathematician See also [ edit ] Dowker Island ,
64-699: 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
96-458: A career as a teacher, but he was persuaded to continue with his education because of his extraordinary mathematical talent. He earned his M.A. from the University of Toronto in 1936 and his Ph.D. from Princeton University in 1938. His dissertation Mapping theorems in non-compact spaces was written under the supervision of Solomon Lefschetz and was published (with additions) in 1947 in
128-431: A conjecture ultimately proven false in a famous 1971 paper by Mary Ellen Rudin . Trajectories 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 ,
160-597: 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 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
192-678: 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
224-411: A particle of mass m {\displaystyle m} , moving in a potential field V {\displaystyle V} . Physically speaking, mass represents inertia , and the field V {\displaystyle V} represents external forces of a particular kind known as "conservative". Given V {\displaystyle V} at every relevant position, there
256-499: 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
288-526: 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 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
320-463: A reasonably good approximation, although if a comet passes close to 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
352-513: A specific person led you to this page, you may wish to change that link by adding the person's given name (s) to the link. Retrieved from " https://en.wikipedia.org/w/index.php?title=Dowker&oldid=1150051014 " Category : Surnames Hidden categories: Articles with short description Short description is different from Wikidata All set index articles Clifford Hugh Dowker Clifford Hugh Dowker ( / ˈ d aʊ k ər / ; March 2, 1912 – October 14, 1982)
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#1732802123059384-409: A trajectory is a sequence ( f k ( x ) ) k ∈ N {\displaystyle (f^{k}(x))_{k\in \mathbb {N} }} of values calculated by 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
416-490: 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
448-411: 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 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
480-556: Is an uninhabited island in Lake Saint Louis, a widening of the Saint Lawrence River south of Montreal Island, Quebec Dowker notation , is mathematical notation Dowker space , is mathematical field of general topology The Haunting of Hewie Dowker , is an Australian film [REDACTED] Surname list This page lists people with the surname Dowker . If an internal link intending to refer to
512-543: 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 the initial vertical speed be v v = v sin ( θ ) {\displaystyle v_{v}=v\sin(\theta )} . It will also be shown that
544-422: 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 ,
576-509: Is the focus of the discipline of ballistics . One of the remarkable achievements of Newtonian mechanics 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
608-464: 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
640-403: 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
672-400: 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
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#1732802123059704-573: The American Journal of Mathematics . After earning his doctorate, Dowker became an instructor at the Western Ontario University for a year. The next year, he worked as an assistant back at Princeton under John von Neumann . During World War II , he worked for the U.S. Air Force , calculating the trajectories of projectiles . He married Yael Naim in 1944. After the war, he was appointed associate professor at
736-471: The Tufts University . Because of Senator Joseph McCarthy 's red scare , he decided to take his family to England shortly thereafter, where he was appointed Reader in applied mathematics at Birkbeck College in 1951. In 1962 he was granted a personal chair, until he retired in 1979. The last years of his life were marked by a long illness, yet he continued working, developing Dowker notation in
768-455: 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 a given initial speed v {\displaystyle v} is obtained when v h = v v {\displaystyle v_{h}=v_{v}} , i.e.
800-405: 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 the point of launch of the projectile. The x {\displaystyle x} -axis is tangent to the ground, and the y {\displaystyle y} axis
832-687: 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
864-425: 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
896-399: The initial angle is 45 ∘ {\displaystyle ^{\circ }} . This range is v 2 / g {\displaystyle v^{2}/g} , and the maximum altitude at the maximum range is v 2 / ( 4 g ) {\displaystyle v^{2}/(4g)} . Assume the motion of the projectile is being measured from
928-402: 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 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
960-479: The weeks before his death. Dowker showed that Čech and Vietoris homology groups coincide, as do the Čech cohomology and Alexander cohomology groups. Along with Morwen Thistlethwaite , he developed Dowker notation , a simple way of describing knots , suitable for computers. His most highly cited article is his 1951 paper in which he introduced the concept of countably paracompact spaces . Dowker conjectured that so-called Dowker spaces could not exist,
992-503: Was a topologist known for his work in point-set topology and also for his contributions in category theory , sheaf theory and knot theory . Clifford Hugh Dowker grew up on a small farm in Western Ontario , Canada. He excelled in mathematics and was paid to teach his math teacher math at his secondary school. He was awarded a scholarship at Western Ontario University , where he got his B.S. in 1933. He wanted to pursue
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1024-401: Was also initiated by Newton in his youth). Over the centuries, countless scientists have contributed to 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
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