What is Trajectory: Definition and 466 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. G

    Maximum Point Blank Range (MPBR) -- Help zeroing my rifle

    I have been attempting to calculate optimal zero for my H&R 30-30. 190g 170g 160g 150g These links are the optimal zeros for my rifle based on each of the four grain weights available to fire thru the rifle. I don't want to have to re-zero my rifle every time I switch between ammo types so I...
  2. P

    Kinematics: Acceleration of a figure skater changing direction

    My guess was simply that as acceleration changes from the north to east direction, the total magnitude change of v is ##v \sqrt 2##. Acceleration is ##\mu g##, so time would be ##\frac {v \sqrt 2} {\mu g}##. This agrees with the textbook solution. What I do not understand is the trajectory...
  3. S

    How to analyse a sequence of vehicle states?

    Hi all, I have to analyse a dataset containing real-world vehicle trajectories and in particular: 1. The trajectories were classified into states in the function of certain vehicle parameters and location (urban roads, country roads, etc.) and each state is characterised by an integer number...
  4. Astronuc

    B Asteroid 2024 BX1 entered Earth's atmosphere over Germany 21 Jan 2024

    Asteroid 2024 BX1 that exploded over Berlin was fastest-spinning space rock ever recorded https://www.livescience.com/space/asteroids/asteroid-that-exploded-over-berlin-was-fastest-spinning-space-rock-ever-recorded Visualisation of the trajectory and impact of asteroid 2024 BX1 on 21 January...
  5. ZeroX4

    Looking for equation/formula for calculating boomerang trajectory based on few variables for game

    So basically i am vary green when it comes to equations/formula Well to the point i don't even know if proper name for what i'm looking for is called equation or formula but let's stick to equation So going as simple as i can I want to throw boomerang creating boomerang like ellipse trajectory...
  6. K

    B Time and right angle traveling twins paradox

    So in scenario in which two ships(A and B) are moving relative to each other, and towards each other; each ship will measure the other to have a velocity. Within special relativity no observer himself moves. Do you agree? If you agree, assume each of the above two ships sends out shuttle along...
  7. L

    Plotting the trajectory of a particle in polar coordinates

    Hi, Unfortunately, I am not quite sure whether I have solved/plotted the following task correctly I started by resolving the expression ##\phi=2 \pi t## to t so that I can represent ##\rho(t)## with ##\rho(\phi)## The vector ##\vec{e}_r## was written in my lecture as follows...
  8. T

    I Help identifying this Tumbling object in a night sky photo from Namibia

    I have received this image from someone in Namibia recently at the Kiripotib Astrofarm. Has anyone an idea what could cause such a strange trajectory?
  9. J

    A Why the James Webb Space Telescope needs propellant

    Orbits around the Lagrange point are not stable, and you need to tweak your trajectory every now and then to remain in that orbit. Simulations of a non-accelerated body give spectacularly weird results: When the propellant runs out, is this what the JWST will do?
  10. R

    A Inquiry Regarding Concept in Time and Celestial Motion

    I am writing to investigate a unique theoretical perspective that explores the relationship between time and the motion of celestial bodies, particularly focusing on Earth's trajectory through space. I posit that time, particularly in the past, is intricately connected to the precise path that a...
  11. P

    I Trajectory collision calculation

    Hello, I ask you for your aid in the solution of the following problem. Please see the attached illustration. Two objects (red and blue) are moving in the vicinity of each other. The red object is moving along a closed circle and the blue object is moving along a line. Our objective is to...
  12. SaintRodriguez

    I Worldline Curve vs Trajectory: What’s the Difference?

    Is a worldline a curve or a trajectory? Why?
  13. C

    Finding the launch angle for a projectile if the range is 3x the max height of the trajectory

    For this problem, The solution is, I don't understand how they got from ##\frac {sin^2\theta_i}{sin2\theta} ## to ##\frac{tan\theta_i}{2}##. Would somebody please show me the immediate steps? I also don't understand is why they say the ignore the trivial case where ##\theta_i = 0~rad##...
  14. L

    I Why do phase trajectories point upwards and downwards in a quadratic potential?

    Hi, I am currently preparing for my exam and have just watched a video about motion in phase space. From minute 4 a quadratic potential is introduced and then from minute 6 minute the phase trajectory. Here are the pictures quadratic potential phase trajectory Regarding phase...
  15. kuruman

    Finding a Parametric Solution for Particle Trajectory in Magnetic Field

    This is a solution to a problem inspired by another thread. It is posted here to separate it from the multiple choice question which was the subject of that thread. A parametric solution for the trajectory can be found quite easily if the motion is modeled as a particle with charge ##q##...
  16. N

    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?
  17. C

    I Looking for a trajectory integrator that also supports cubic potential

    The context: I created an educational resource, a set of interactive diagrams that allow the user to see how Hamilton's stationary action arrrives at the true trajectory. There is a diagram for each of the following three cases: - Uniform force, hence the potential increases linear with...
  18. N

    I Solving for the trajectory of the center of mass

    I'm working on the physics engine component of a game engine I'm building, and I need some guidance with this particular situation. Consider a square with mass M that is free to translate in the xy plane and free to rotate about any axis perpendicular to the page (Fig. 1) If a linear impulse J...
  19. E

    B Can nuclear bomb(s) change trajectory of small black hole?

    A 4 ft diameter black hole (1.5 to 2 times the mass of Saturn) is headed toward Earth. Will Earth be able to defend itself using missiles with nuclear bombs? How to calculate such a problem?
  20. A

    Trajectory of an electron traveling near a current-carrying wire

    B equals 50*10^-7 T (at first instance) Fm equals 8*10^-20 N (at first instance) I know Fm is perpendicular to the velocity, and I know the estimation of the trajectory (somewhat similar to the curve y=lnx). Since I think vertical velocity will be constant, only changing the x component, I...
  21. C

    Space-like trajectory in Schwarzschild spacetime

    I'm not sure how to approach this question. So I start off with the fact the path taken is space-like, $$ds^2>0$$ Input the Schwarzschild metric, $$−(1−\frac{2GM}{r})dt^2+(1−\frac{2GM}{r})^{−1}dr^2>0$$ Where I assume the mass doesn't move in angular direction. How should I continue?
  22. M

    Electron Trajectory: Stuck on a difficult problem....

    I am very confused on how to go about with this question. The only thing I've tried so far is drawing the fbd of the electron and because of the plates the force applied would cancel which makes it centered between the two plates. As of now that is the only thing I understand, I am not sure how...
  23. O

    B Can Unlimited Energy Launch Objects into Orbit Using Accelerator Technology?

    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...
  24. P

    I Shooting an electron past a positive nucleus (trajectory)

    An electron is shot horizontally. There is a proton located somewhere else, but not in the horizontal path of the electron. Is there a distance of closest approach, and how do you derive it? A physical explanation would be appreciated too.
  25. Z

    Comp Sci Solving the ballistic trajectory equation for Launch Angle

    In a process of writing a game. Effectively need to know how to angle the barrel for the projectile to hit the selected target. So for the equation y = h + x * tan(α) - g * x² / 2 * V₀² * cos²(α) Everything except α is known. Could anyone more wise in the ways of science than me help me solve...
  26. N

    Displacement and distance when particle is moving in curved trajectory

    While solving question 1.13(see the attachment) from Irodov, I was doing this: $$\int_{0}^{\tau}(\vec{v}-ucos\theta) dt=l$$, and $$\int_{0}^{\tau}\vec{v}cos\theta dt=u\tau$$. Solving this gave the answer. However, while solving these 2 equations, I only used the magnitude of ##\vec{v}##, and...
  27. greg_rack

    Circular trajectory traveled by a charged particle in a magnetic field

    The Lorentz's force acting on a charged particle perpendicularly "hitting" a magnetic field will be directed upwards, and generally directed towards the center of the circumference traveled by this particle, and so will cause a centripetal acceleration to keep it in a circular motion. By...
  28. Andrei0408

    Understanding the Trajectory Equation: X^2+Y^2=5

    So I've done the first part of the problem, I just need to understand the x^2+y^2=5 part. I believe in order to get to this (I think this is called the trajectory equation, but please correct me if I'm wrong), we wrote sin(3t) = x/5 and cos(3t) = y/5, we raised each of these to the power of 2...
  29. J

    Express this trajectory in terms of proper time

    The object moves solely on the $x$-axis, hence I calculated its speed to be $v_x = \frac{dx}{dt} = \frac{c \kappa t}{\sqrt(1+\kappa ^{2} t^2$ Because its speed is not constant, I suppose the Lorentz factor $\gamma = \gamma (t)$, and by plugging in the velocity, I obtain $\gamma = \sqrt(1+ \kappa...
  30. JackFyre

    Projectile Motion: Achieving a Semicircular Trajectory without Calculus

    Hey Folks! I've got a longish one- At what angle must you release a projectile to achieve a semicircular trajectory(neglecting air-resistance)? Would the initial release velocity matter? and would the same criteria hold true if the value of g were different(on the moon for example)? and is there...
  31. burian

    Trajectory of a mass element in a string overhanging a pulley

    Hello everyone, this is a thought experiment I made, it involves two ropes being pulled of a pulley similar to an atwood machine, and a block attached to the two pulleys at the other end. The ropes are being pulled at a constant velocity $ U$ from the end away from the block and the block rises...
  32. LCSphysicist

    Trajectory in a non inertial frame

    Well, make the center of the polar coordinates at the center of the turntable, so put r along the ro initial. I am well known that to someone who is rest on the turntable, the equations will be the follow: dr/dt = ro - vt If the turntable route with angular velocity w, dtheta/dt = -wt We...
  33. CraterHater

    Help with adding air resistance into my projectile trajectory function

    Hey, I am working on a video game in which there will be archers who have the ability to shoot at enemies. My game is two dimensional and I am trying to calculate the angle at which the archer, given an initial velocity, has to shoot in order to hit the target perfectly. I came up with the...
  34. 1

    Help with the trajectory of a particle in a magnetic field

    Hello, sorry for this stupid question but I am getting confused with equations and I need some help. My problem is the following: I have a dipole with known magnetic field (B=0.234T), it is 110.6 mm long and 89 mm wide (mechanical dimensions). I want to calculate deflection of particles...
  35. M

    Electron with circular trajectory in a magnetic field

    I'm not sure how I'm able to calculate the velocity of the particle using the formula without knowing the force exerted on it. Also, I don't understand why the question also provides the mass of the electron.
  36. 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...
  37. L

    Deriving an equation for theta via the Projectile Motion Trajectory Eq

    Perhaps I should ask this question in the math section of these forums. But, I'm stumped and I don't think this equation is solvable? Find theta given all the other variables:
  38. F

    Why is my simulation of projectile trajectory with air resistance wrong?

    Hey! This started very harmless... A friend and I were throwing stones in a lake. Mine didn't get very far, he was teasing me "What was the ideal angle again?". Of course, I know it should be 45°. I replied in jest: "That's because I'm considering air resistance!" Then we had a discussion what...
  39. 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...
  40. 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...
  41. Benjamin_harsh

    What is the proper formula for ##Y_{max}## for this trajectory?

    Time it takes from A to travel to B; ##T = \large\frac{Vsinθ}{g}## Max height object can reach, ##h = \large\frac{V^{2} sin⁡2θ}{g}## Time it takes for for B to travel to C; ##T = \large\sqrt\frac{2y_{max}}{g}## What is the proper formula for ##Y_{max}## for this trajectory?
  42. Demystifier

    I What is the current experimental evidence for quantum trajectory theory?

    [Moderator's note: Spin off from previous thread due to topic change.] Recent experimental confirmation of the quantum trajectory theory: https://www.quantamagazine.org/how-quantum-trajectory-theory-lets-physicists-understand-whats-going-on-during-wave-function-collapse-20190703...
  43. Avatrin

    Showing rotational trajectory with Python

    Hi I have a list of 3D angular velocities (a numerical solution to an ODE). I want to show the trajectory this rotation would cause by mapping it out on the unit sphere. How can I go about doing that? What is the best way to approach this?
  44. Avatrin

    Python Plotting a 2d trajectory with Matplotlib

    Hi I have a 2d trajectory of a center of mass which I have saved in an array such that r=[[x(t_1),y(t_1)],[x(t_2),y(t_2)],...,[x(t_n),y(t_n)]] where x is x coordinate of the position and y is the y-coordinate of the position. I have tried this: xx = [t[0] for t in r] xy = [t[1] for t in r]...
  45. A

    Solving 2nd order ODE in order to get equation for Orbital Trajectory

    I want to solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u(\theta)##, where ##\frac{GM}{h^2}=constant##. The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients: ##\frac{du^2}{d\theta ^2}+u=0## which...
  46. A

    Trajectory of charged particles in a magnetic field?

    A charge is entering magnetic field perpendicularly! Griffiths says it motion will be circular! But it is accelerated so it must radiate energy and it's motion should be spiral inward?
  47. Badgun

    Impact of several variables on resulting projectile motion trajectory

    I was told to generate these variables (m, C, alpha, wind velocity) normally distributed and compare the random data with the result and then tell, which of the variables has the most impact. Here I am stuck, tried to compare variances, kurtosis and skewness of the data (the original variables...