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# Acceleration differential eqn for distance. watch

1. Hi,

So I'm quite confused about the following Q:

A particle's motion is governed by the differential equation

where

with

and

v decreases as the particle moves, until v=0 after it has gone a distance D.

I've tried integrating w.r.t t but nothing seems to be working out.

The distance is supposed to be int terms of a log by the way.

Any tips?

Thanks!
2. (Original post by golden ratio)
Any tips?
One standard technique is to rewrite dv/dt as dv/dx dx/dt = v dv/dx

And then separate variables and integrate wrt v and x, not t.
3. (Original post by ghostwalker)
One standard technique is to rewrite dv/dt as dv/dx dx/dt = v dv/dx

And then separate variables and integrate wrt v and x, not t.
So

and

But now I'm unsure what to do in order to be able to apply the IC

Or is it

?

Thanks again!
4. (Original post by golden ratio)
Or is it

This. Might be easier to leave the minus's on the right.
5. (Original post by ghostwalker)
This. Might be easier to leave the minus's on the right.
I thought so, but then I'm unsure as to how to get t back after I integrate (to apply the IC), because

6. (Original post by golden ratio)
I thought so, but then I'm unsure as to how to get t back after I integrate (to apply the IC), because

If you find v in terms of x then use v = dx/dt you will end up with another (hopefully simple) DE to solve

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