Trajectory Definition and 39 Discussions

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.
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








{\displaystyle (f^{k}(x))_{k\in \mathbb {N} }}
of values calculated by the iterated application of a mapping


{\displaystyle f}
to an element


{\displaystyle x}
of its source.

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  1. NODARman

    Change of ellipse while accelerating the rocket

    When the rocket accelerates in space does its trajectory which is an ellipse change in size and not the focal points because the Earth is still in one of two and also the current height doesn't increase, right?
  2. A

    Trajectory of a charged particle in the magnetic field (variable)

  3. O

    B Untraditional Space Travel

    If an infinite amount of energy were available to create the lift mechanism for a space launch. What would be required to fire a 200lb object into low Earth orbit(160km) after speeding it up in a way similar to how the large hadron collider speeds up a particle. Assuming the launch vehicle...
  4. J

    Quaternions and Direction Cosine Matrix changing in time

    I've already posted this question on the mathematics website of stack exchange, but I've received more help here in the past so will share it here as well. I am developing a tool for missile trajectory (currently without guidance). One issue is that the aerodynamic equations on the missile are...
  5. SamRoss

    I How can the stress tensor be non-zero where there is no matter?

    You're on Earth. You throw a ball and watch its trajectory. It's curved. That's because the Earth is curving space-time at every point along the trajectory. But the Earth itself is not present along the trajectory - there is no matter along the trajectory (let's ignore the air and any radiation...
  6. K

    Plotting a rotating ellipse trajectory (the Moon orbiting the Earth in 2D)

    I'm tasked with drawing the trajectory of the Moon around the Earth (in 2D), taking into account the fact that the trajectory also undergoes precession, so the elliptical orbit rotates about it's center (I think it should rotate around the Earth-Moon barycenter, but for the first step I...
  7. D

    I Why do rockets turn horizontally so soon after launch?

    My 12 year old asked me this question. I have a MS in Mechanical Engineering, so I can usually answer his physics questions, but this one stumped me. When lifting off, why do most rockets turn close to horizontal almost immediately? Of course we know they need mostly horizontal speed to...
  8. S

    I Calculate Voyager trajectory using JPL data

    Hey everyone, I've recently programmed an animated simulation of the main elements (core planets and Sun) of our solar system: by using the initial coordinates from the JPL database, and then calculating the combined gravitational perturbations and the relativistic effects from the Sun...
  9. T

    Trajectory of a Projectile that must clear an obstruction

    I am trying to solve for Vy with given Vx. I know how to do this for the BLUE trajectory line, but I do not know how to solve this if there is an obstruction in the way (like the gray wall). My givens are: X1 X2 Y1 Y2 Vx Gravity Do I need to solve for the blue trajectory then adjust that for...
  10. kandelabr

    Trajectory with minimum acceleration

    Currently design of turbomachinery (impellers/turbines) is more a form of art than an engineering process. One has to guess a bunch of parameters and check if they are right in much later stages of design. I was thinking about designing the other way: we know the initial and the final velocity...
  11. Rednecknav

    I Space Trash

    How much energy needs to be be expended to throw a non operating satellite (or other space trash) out of orbit (L4 or L5) or L2 into deeper space... thus minimizing space debris in Earth orbit. I could only find formulas for getting an object INTO orbit... Background of question: we were...
  12. V

    Calculating the trajectory of an object into a force field

    How can I calculate the trajectory of an object knowing the initial conditions and the function F(x,y)? For example, a object orbiting a planet or two electric charges orbiting each other.
  13. B

    Velocities when generating an Archimedean spiral trajectory

    hey, I just want to know, if I am to send a velocity commands to generate a spiral trajectory, What would be these velocities (angular and linear)?? Thanks in advance
  14. H

    Equation for Brownian Motion Trajectory

    I am trying to understand how one can simulate the trajectory of a Brownian particle as a function of time. I am only able to do it with the assumption that I can simply generate random values of x and then take the cumulative sums of these values to get the trajectory of the Brownian particle...
  15. peasqueeze

    Modified Cycloid motion of charged particle

    I need a charged particle trajectory in crossed E and B that is a cycloid in the x-y plane. This part is simple enough, but I want the trajectory to be parabolic in the x-z plane. The images will help explain what I need. The left image shows the shape I want to achieve, and the right image...
  16. I

    I Name for electron's trajectory outside of the orbit?

    I wonder if this question will make sense, but I will do my best shot: As the electron orbits around the nucleus, I imagine there are moments when it is too far away, so it starts to be "pulled toward" the nucleus. If it gets too close, it will be "pushed away". If it is in the orbital area...
  17. J

    What is known about helical flight paths in bullets?

    I have a physics degree, I'm now retired and have taken up long range shooting as a hobby. When shooting at 600 yards, my spotters have noticed the contrail of my bullet follows a helical trajectory near the target. I shoot a 243 (6mm), 105 grain target bullet made by Berger at a muzzle...
  18. B

    I Identical particles distinguished by their trajectory?

    Hello. It is said that if we exchange two electrons, we can't tell which is which. Identical mass, charge, etc. So if I hold two electrons, one in each hand, and someone switched them, I wouldn't be able to know. But one way to distinguish particles is their trajectory. If I have a very long 1D...
  19. N

    I 4th order Runge-Kutta with system of coupled 2nd order ode motion equations

    MX''=Fn(cosΦ−usinΦ) MZ''=Fn(sinΦ+ucosΦ)−Mg MΦ''=Fn(Bxx+uBz) I tried using Runge-Kutta methods to approximate motion equations in MATLAB but it turn out wrong. I also tired finding and researching forums and web for solution but to no avail. Fn,M,θ,u is constant fn/M = 0.866 it seems that i...
  20. arpon

    Determining shortest possible "time" to reach destination

    Homework Statement Two towns A and B, are situated directly opposite to each other on the banks of a river whose width is 8 miles and which flows at speed of 4 mi/hr. A man located at A wishes to reach town C which is 6 miles upstream from and on the same side of the river as town B. If his...
  21. D

    Comp Sci [JAVA GUI] Counting the trajectory of a projectile not shown

    Hello PFers, I'm trying to make a program in java which process user's position, initial velocity of the projectile and the elevation of the projectile and show the trajectory in GUI. The language is in java. The problem is, the trajectory won't show. It is supposed to show the player's...
  22. C

    What is the Lift Coefficient of a Basketball?

    How can I find an equation to calculate the lift coefficient on a basketball at different launch velocities? I looked at different papers explaining the physics of basketball throws but all i could find was an equation for the drag coefficient: -mv +1/2 ρ A Cd v^2 = mv
  23. B

    Projectile Motion spring launch

    Homework Statement I am trying to find the force needed from a spring to launch a mass of 800g to a length of 0.875m. This launch would be at the optimum angle of 45º. I am aware of all projectile motion equations but am having difficulty linking them to force and thus solving the problem...
  24. O

    2D trajectory calculation with constant force

    Hello folks, I need some physics help. Some background to get us on the same rails :) I am developing a 2D game in my spare time. Player controls an object (aka spaceship) in open space (no gravitation influence) by selecting a point in space. A constant force which is always pointed at the...
  25. MattRob

    The Validity of the Schwarzschild Metric Inside of a Real BH

    So, I've been reading through "Exploring Black Holes: Introduction to General Relativity" by Wheeler and Taylor, and I've had some ideas I wanted to pursue and do some research in regarding trajectories within the event horizon. In this, I'd like to have the mathematical tools to investigate...
  26. Xsnac

    2 Perpendicular oscillations

    Homework Statement I have 2 perpendicular oscilations and I have to find the trajectory equation. $$x=A\cos\omega t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)$$ $$y=B\cos(\omega t+\Delta\phi)$$ Homework Equations $$\cos (x+y) =\cos x\cos y -\sin x\sin y$$ $$\cos^{2} x+\sin^{2} x =1$$...
  27. P

    Trajectory opimization: Fast preview algorithm

    Hello everyone, I have been interested in trajectory optimization for a while now and I have read a few papers on that topic and bought the book "Spacecraft trajectory optimization" from Cambridge University Press and want to start programming with the goal to optimize a trajectory in a...
  28. W

    Calculate the Pitch of a projectile with moving destination

    Hi all. I am not quite sure if this is the right forum to place this question but for me there are several forums that would fit with my situation :D I am making a game where I have to throw/shoot a projectile through a 3 dimensional space which has to land on a specific object. There are...
  29. R

    Trajectory of a turning particle

    In this problem, I need to find the trajectory of a particle (as a function of time) which moves at a speed 's' but also turns at an increasing rate; angular acceleration α. The trajectory looks like a spiral which converges to a point. The particle has an initial position vector p and a...
  30. AdityaDev

    Trajectory of centre of mass

    Homework Statement Lower surface of a plank is rough and is initially at rest on a horizontal floor. It's upper surface is smooth and has a smooth hemisphere place over it through a light string as shown. Find the trajectory of centre of mass of the sphere after the string is burnt. Homework...
  31. A

    Trajectory lab

    Homework Statement So I am doing a trajectory lab. I take a ball and release it from the ramp that is on the table and then measure the distance. The thing is when I graph the angle of the ramp vs the distance the ball has fallen I get parabola. And the correlation of parabola is 0 Homework...
  32. D

    Trajectory of dart

    I don't quite get, where does the V0x=v * d/(sqrt(d^2+(h-y0)^2) come from? also, what does (h-y0) equal to? t/v0y?
  33. Archit

    Help me calculate the speed of the basketball

    Here is the picture for you to understand it better. Please help me calculate the velocity required for the ball. I have already tried it with the formula: displacement = initial velocity(time) + (acceleration)(time2)/2 but cannot since I do not know the initial velocity and the time Please...
  34. G

    What's the distance a sphere travels from inclined plane?

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > So the problem that I have been assigned has formulas of rotational energy, momentum, trajectories, inertia, and inclined planes. A solid sphere is rolling down an inclined plane (that is placed...
  35. J

    How can a wave have a continuously changing trajectory?

    Consider the release of a single photon or electron into a vacuum. The particle will have an initial trajectory. How is it possible for this event to give rise to a wave that has a continuously changing trajectory?
  36. O

    Calculating trajectory for a throw (Game-programming)

    Hello, and +1 for this great forum! I have encountered a small problem in my way of creating a small game. I have a snowball that is being thrown by the player. The dynamic variables from the input of the player are: Force and angle (between 90 : -90 degrees). At the moment I apply the force...
  37. SalfordPhysics

    Comp Sci Fortran - equation of motion, astronomical units

    Homework Statement Euler method : Plot the trajectory of a body moving under the influence of the suns gravity from initial conditions x=1, y=0, vx=0, vy=1. My trouble is figuring out my function. Homework Equations d2r / dt2 = -r/r3 The Attempt at a Solution What I have been doing...
  38. K

    Trajectory of planet

    Hi all. This may be a stupid question to some.. But please do answer it. Why do planets move in an elliptical path? Why only ellipse? Or even a circular path for approximation? Why not a path which has corners? Like an octagon or a square?
  39. C

    Look of trajectory of a material point

    Homework Statement A body is moving, with the following data (these are all dependencies of the radius vector components respective to time): x(t)=At2, A>0; y(t)=Bt, B>0, z(t)=0. How does the trajectory of this movement look? Homework Equations r=x(t)i+y(t)j+z(t)k Velocity=dr/dt The Attempt at...