# Kinematics of movement subject to a variable acceleration

• MatinSAR
In summary, someone can guide someone how can they find time by using a search engine like Google. This is a common approach to solving a problem involving acceleration that changes with position.

#### MatinSAR

Homework Statement
Can someone help me to find How long does it take for a moving object to travel a distance d with an acceleration varying with position?
Relevant Equations
a=GM/r^2
G and M are constant and this acceleration is due to the force of gravity between two bodies.
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.

MatinSAR said:
I don't have any idea.
There is a well-known search program called Google.

 inadvertently inserted the wrong 'link'. This was what I intended: https://en.wikipedia.org/wiki/Two-body_problem

Inspired by the ##GM\over r^2## (## \vec a = -{GM\over r^3}\,\vec r## ## ! )

also see Kepler orbit

##\ ##

Last edited:
topsquark, jack action and berkeman
Are you familiar with how to derive the velocity and position using integrals to accommodate the varying acceleration?

topsquark and MatinSAR
BvU said:
There is a well-known search program called Google.
Thank you ...
I have searched but I didn't find anything.
All I have founded was related to acceleration that changes with time.
This type(acceleration that changes with time) is easy but this problem has acceleration that changes with position.

berkeman said:
Are you familiar with how to derive the velocity and position using integrals to accommodate the varying acceleration?
Yes.
but this question is different. The acceleration changes tith position.

MatinSAR said:
Thank you ...
I have searched but I didn't find anything.
All I have founded was related to acceleration that changes with time.
This type(acceleration that changes with time) is easy but this problem has acceleration that changes with position.
It is possible to find the solution to this online, but in the spirit of the HH forum here what have you tried so far? How much of this can you do yourself? Can you use a conservation law?

topsquark
PeroK said:
It is possible to find the solution to this online, but in the spirit of the HH forum here what have you tried so far? How much of this can you do yourself? Can you use a conservation law?
Can you share a link ?! I didn't find any thing related.
and It's not a question. This is a part of a question in Classical Dynamics which I am trying to solve and I don't know how can I find time.

We can use conservation law.

I get that, but there are homework helping rules here.

topsquark
MatinSAR said:
Thank you ...
I have searched but I didn't find anything.
All I have founded was related to acceleration that changes with time.
This type(acceleration that changes with time) is easy but this problem has acceleration that changes with position.
I literally used your words in DDG and got this as a first result:

With the same words in Google, this was the second result (but the results may be different for someone else):

berkeman, MatinSAR and PeroK
PeroK said:
I get that, but there are homework helping rules here.
I didn't ask anyone to solve my problem. I didn't even send the main problem which was about gravitational force !
I have asked how can I find time when acceleration changes with position and it's not a homework.
Thank you for your time anyway.

jack action said:
I literally used your words in DDG and got this as a first result:
I have forgotten about other search engines !
Thank you I will read them.

MatinSAR said:
how can I find time when acceleration changes with position

Given acceleration as a function of displacement, the usual approach is to eliminate time:
##\dot v= f(x)=v\frac{dv}{dx}##
##\int f(x).dx=\int v.dv=\frac 12v^2##
This is just the energy conservation equation with mass cancelled.
##t=\int\frac{dx}{g(x)}##

MatinSAR
MatinSAR said:
I have asked how can I find time when acceleration changes with position
When I am attacking a problem that involves acceleration changing with position, I like to think instead about a fixed mass under a force that changes with position. Now the familiar concepts of work and energy come into play. The change in kinetic energy of the mass will be the integral of force over distance.

This does not get you all the way to a result. But with kinetic energy as a function of distance and a known mass, one can obtain velocity as a function of distance. Or invert velocity and get incremental elapsed time as a function of incremental traversed distance.

Which is what @haruspex just finished saying, but without the algebra.

MatinSAR
MatinSAR said:
and It's not a question. This is a part of a question in Classical Dynamics which I am trying to solve and I don't know how can I find time.
MatinSAR said:
I didn't ask anyone to solve my problem. I didn't even send the main problem which was about gravitational force !
I have asked how can I find time when acceleration changes with position and it's not a homework.
Please calm down. We treat all schoolwork-type questions the same here at PF; we are treating your question the same.