Acceleration as a function of position, and time taken to travel a distance

[jmz34]

I've been doing a problem that requires me to find the time taken to travel a certain distance if I know the initial acceleration of a body at the starting position and its initial velocity (starts from rest). The acceleration is a function of position a=-GM/(Ro^2).

So say a body is released from rest a distance Ro from a point mass, and it's initial acceleration is that stated above- how would I go about finding the time taken to travel this distance (from Ro to the origin.


Thanks.

 

[tiny-tim]

hi jmz34!

so r'' = -GM/r² ?

multiply both sides by r' …

r'r'' = -GMr'/r²

… and integrate ​

 

[jmz34]

hi jmz34!

so r'' = -GM/r² ?

multiply both sides by r' …

r'r'' = -GMr'/r²

… and integrate ​

Using this method I did this:

d/dt(0.5*(r')^2)=(-GM/r^2)r'

then integrated once and simplified to get

(dr/dt)^2=2GM/r

solving this for t gives:

t=(1/3)*SQRT(2/GM)*Ro^(3/2)

If you could have a quick look at my method I'd be very grateful.

Thanks alot.

 

[tiny-tim]

Using this method I did this:

d/dt(0.5*(r')^2)=(-GM/r^2)r'

then integrated once and simplified to get

(dr/dt)^2=2GM/r.

yes, but after that i get a bit lost …

you seem to have lost r completely, and you don't have a constant of integration

 

[rwisz]

I would approach this problem by using the equations of motion and the principles of calculus. First, I would use the equation of motion for constant acceleration, which is s = ut + 1/2at^2, where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time taken. In this case, the initial velocity is 0 since the body starts from rest.

Next, I would use the given acceleration formula, a = -GM/(Ro^2), and substitute it into the equation. This would give me s = 1/2(-GM/(Ro^2))t^2.

To solve for t, I would use calculus and take the derivative of the distance equation with respect to time, which would give me the velocity equation, v = at. Then, I would set the velocity equal to the initial velocity (0) and solve for t. This would give me t = sqrt(2Ro^3/GM).

Therefore, the time taken to travel from Ro to the origin would be sqrt(2Ro^3/GM). This calculation assumes that the body is traveling in a straight line towards the origin and that there are no other external forces acting on it. I hope this helps to solve your problem.