# What is Variable acceleration: Definition and 36 Discussions

In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law of motion directly to such a system. Instead, the time dependence of the mass m can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system. The general equation of variable-mass motion is written as

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{\displaystyle \mathbf {F} _{\mathrm {ext} }+\mathbf {v} _{\mathrm {rel} }{\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}
where Fext is the net external force on the body, vrel is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body, and v is the velocity of the body. In astrodynamics, which deals with the mechanics of rockets, the term vrel is often called the effective exhaust velocity and denoted ve.

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1. ### Kinematics of movement subject to a variable acceleration

Can someone guide me how can I find time ?! I don't have any idea. This is a part of a question in Classical Dynamics.

3. ### Kinematics Motion with Variable Acceleration

The displacement of the particle is; s= ∫v dt s= ∫4-3t^2 dt s=4t-t^3+c When the particle is at the Origin, t=0; 0=4(0)-(0)^3+c c=0 So this becomes; s=4t-t^3 The particle next passes through the origin when; 4t-t^3=0 Factor out the common term -t; -t(t^2-4)=0 Rewrite t^2-4 as t^2-2^2 and factor...
4. ### Modifying Newton's Kinematic Equations for High Altitude

I recently watched a video on youtube where a guy fires a 50 cal. rifle straight up into the air and measures the time of flight at about 100s. It got me thinking about what altitude the round reached. So I used the kinematics equations and obtained a value of 12.25km. Amazing, but I wondered...
5. ### Pendulum time at any given angel θ, 0<=θ<=π/2

I got stucked with a solution of a problem. I have a pendulum of r radius which is in G gravitational field. So the working acceleration on the bob is a function of the angle. That is Gcosθ. The mass of the bob = m The radius of the string of the pendulum = r Gravitational force is G. So the...
6. ### Motion under variable acceleration experiment

Used a mass attached to a dynamic cart via pulley to examine motion under variable acceleration. As the mass was increased, my calculated friction force shows a decrease. I know that friction should increase with larger mass. Any suggestions as to why my calculations show a decrease? Used the...
7. ### Real world applications of Jerk and Cons. Acceleration?

I did some light searching for Jerk and didn't find much, maybe I can be directed there if my question has been asked. I was tasked to find some real world differences between the two in my Dynamics class. It's not really homework, so I didn't post it there. I'm looking for a discussion from...
8. ### Rectilinear motion of two attracting masses

Homework Statement Consider two masses of variable magnitude (M m) that are separated by a distance ( r ). Both masses are free to move. Calculate dr/dt. Homework Equations (See below) The Attempt at a Solution F = GMm / r2 a = Gm / r2 let k = Gm da/dr = -2kr-3 dr/da = (-2k)-1 r3 dr/da *...
9. ### Calculating velocity with variable acceleration

This is not so much coursework as it is a thought experiment that has been bugging me for a while now. 1. Homework Statement Find the equation for velocity either as a function of distance (x) or time (t). I have an object whose acceleration is given by the rational function a(x) =...
10. ### Boat slowing down with variable acceleration

A motor boat of mass m moves on the surface of a lake at a speed0v. At the moment 0t= the engine is shut down. Assuming the resistance of water to be proportional to the speed of the boat, F=−rv, find a) the time interval after which the boat stops; b) the speed of the boat as a function of...
11. ### Variable acceleration and terminal velocity

Homework Statement A cube is falling from a height of 200 m (initial velocity =0) , it reaches it's terminal velocity . Given that :its mass is 100 kg , cross sectional area 40 cm2 and coefficient of drag 2. Putting only weight and drag in consideration , calculate its terminal velocity and...
12. ### Dynamics help: variable acceleration and angular velocity

I have to do these 2 problems for homework, but I can't figure them out. Any help would be appreciated. 1. The motorcycle is traveling at 1m/s when it is at A(the origin). If the speed is then increased at \dot{V} = .1 m/s^2, determine its speed and acceleration at the instant t = 5 seconds...
13. ### Finding final velocity with variable acceleration due to air resistance

Homework Statement Hello! This is my first time posting here but I have been viewing these forums for help in my physics class for a while. Lately I have been stuck on one single problem that I haven’t been able to find any help with anywhere online. Any who, here is the problem: a cat...
14. ### I think I get it now. Thanks for your help. You're welcome, happy to help!

Homework Statement An object starts from position X = 0 and moves along a straight line with its position given by x = 4t^2 - t^3 where t is the time taken to reach position x. Find: i) The time at which the velocity is zero ii) The time at which the acceleration is zero iii) The average...
15. ### Calculating Velocity and Acceleration for Variable Motion

An object starts from position X = 0 and moves along a straight line with its position given by; X = 20t - 5t^3 where t is the time taken to reach position X. Find; i) The time at which the velocity is zero ii) The time at which the acceleration is zero iii) The average velocity over the...
16. ### Finding displacement from variable acceleration

Hello, I've been staring at this question for an hour with no luck.. if anyone could point me in the right direction that would greatly appreciated! Homework Statement a = acceleration v = velocity x = displacement m and n are constants vt = terminal velocity Show that after...
17. ### Variable Acceleration - Clueless

Homework Statement The problem is labeled Variable Acceleration. The acceleration of a marble in a certain fluid is proportional to the speed of the marble squared and is given (in SI units) by the expression Ax = -3.00 Vx^2. If the marble enters this fluid with a speed of 1.50 m/s, how long...
18. ### Vaidity of rotational formulae for variable acceleration

Homework Statement Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each case. (a) v = rω; (b) atan = r\alpha; (c) ω = ω0 + \alphat; (d) arad = rω2; (e) K = 0.5Iω2. Homework Equations The Attempt at a...
19. ### Question about frictionless motion with variable acceleration.

1. Person is climbing a mountain with rope attached to rock on a flat surface above for support. The slope has an incline angle of 50 degrees. The rock is on frictionless ground, and the mass of the rope can be ignored. The person (100 kg) is 3 meters from the flat surface, the rock (700 kg) is...
20. ### Equation for motion with variable acceleration

I'm stuck on the solution of a problem. I'm looking for the general equation for the motion of a body with respect to time as follows: A mass slides down a frictionless slope. The slope is not linear, but has the function y = -x^2 so that as the value of x increases, the acceleration...
21. ### Distance covered under variable acceleration

Is it possible to find the distance covered in a certain time when acceleration depends on velocity, such as an electric-powered vehicle might do?? Acceleration := K*velocity + Const
22. ### Constant Power, Variable Acceleration

Homework Statement A car of mass m accelerates from rest along a level straight track, not at constant acceleration, but with constant engine power, P. Assume that air resistance is negligible. a) Find the car's velocity as a function of time b) A second car starts from rest alongside...
23. ### Variable acceleration with one premise

Homework Statement A particle's acceleration is given by the equation a(t) = 2t + 1 m/seg^2. If its velocity at t = 2 equals 5 m/seg, how much does its displacement change from t = 2 to t = 4? Homework Equations The Attempt at a Solution I thought I could integrate the...
24. ### Solving for Constant Acceleration and Time in 1D Motion Problem

Hi Guys, I'm a programmer working on a hobby project (physics simulation), and came across this forum while looking for ideas related to a 1 dimensional motion problem I have. Hopefully someone can help me out (or at least point me in the right direction!). =) Homework Statement An...
25. ### Calculating Variable Acceleration for UAV Recovery

Im trying to calculate the variable acceleration of UAV being recovered on a patrol boat. I have the inital velocity, the final velocity, the mass. In the long run i need this acceleration to calculate the recovery distance. I can iterate a time step for the recovery so time is also known. can...
26. ### Work, Energy and Power - Variable Acceleration

The question is about Mechanics (so a mix of maths and physics) Homework Statement A car of mass 900kg moves along a straight horizontal road with its engine working at a constant rate of 20kW. Its speed at A is 10m/s, Assuming that there is no resistance to motion, calculate the time taken...
27. ### Kinematics Problem involving variable acceleration

Homework Statement What if the acceleration is not constant? A particle starts from the origin with velocity 8 i m/s at t = 0 and moves in the xy plane with a varying acceleration given by vector a = (7√t j) m/s2, where t is in s. Answer the following using t as t as necessary. Determine...
28. ### Aircraft landing (drag -> variable acceleration?)

1. An F-15 touches down with a velocity of 100m/s. The only decelerating force acting on it is the drag, given by- How far does the aircraft move before it stops?2. D = 1/2 x Cd x ρ x A x v^2 (add in your own values for Cd, ρ, and A) a = (-D/m)3. Since the D depends on the...
29. ### Variable acceleration question

hi all, i think this should be simple but i just can't figuring it out i basically want to move a particle at rest from point A to point B by applying a force, F(t). the particle should be at point B when t = T knowing at that F = m * a(t), i m assuming m = 1, which means F(t) = a(t)...
30. ### A fixed cannon is to fire a projectile at a tank

1. Problem Statement: A fixed cannon is to fire a projectile at a tank moving toward the cannon at 40 mph. If the cannon is to fire at the moment the tank is 8 miles from the cannon, and the muzzle speed of the projectile is 1000 mph what is the correct equation to determine the firing angle?2...
31. ### Variable acceleration, time, distance

Homework Statement A particle moves along a straight line. At t=0 the acceleration is k(t+2), initial velocity = 0 and x=5m. a) Write down the position of the particle as a function of time. b) At what point in time does the particle reach the velocity 16m/s, and where is the particle at...
32. ### Modeling Time to Cover Distance with variable acceleration

Greetings all. I am attempting to write a program that will calculate how long a spaceship will take to move from Point A to Point B. This is actually an attempt to model my favorite computer game, so it isn’t 100% accurate in comparison to the real world. The developers introduced an element of...
33. ### Variable acceleration terminal velocity

4 conditions - initial velocity = 1000 m/s - height from Earth = 100 km - total mass = 3000 kg - rocket capabilities (thrust) = 100 kg/s at MACH 5 i was told that to use viscous and laminar drag as two options and to keep mass constant (to simplify things a little). the spaceship is...
34. ### Need HELP equation of motion with variable acceleration

Homework Statement A particle is dropped from rest, at the surface, into a tank containing oil The acceleration of the particle in the oil is a = g – kv where g is the gravitational acceleration and –kv being denoted by the resistance put on the particle by the oil. Solve for x as a...
35. ### Equation of motion with variable acceleration

I have recently been working on a project regarding black holes and the spaghettification aspect of it interests me quite a bit. So, I have set out to try to derive some mathematical descriptions of the geometry of the object being spaghettify. I have spent a few hours (uncessfully) trying to...
36. ### Variable acceleration problem.

A particle of mass m is released from rest at a distance b from a fixed origin of force that attracts the particle according to the inverse square law: F = -kx^-2 Show that the time required for the particle to reach the origin is: [pi](mb^3/8k)^1/2 I have no idea where the pi came...