Finding displacement from variable acceleration

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The discussion revolves around finding displacement from a variable acceleration equation given by a = m - nv². The user is struggling to derive the displacement equation x = -0.5ln(1 - (v/vt)²) and has attempted to integrate using acceleration and velocity relationships without success. A suggestion was made to use the equation vdv/(m - nv²) = dx to facilitate the integration process. The conversation highlights the challenges of integrating variable acceleration and the need for clearer steps in the solution. The thread emphasizes the importance of proper substitution and integration techniques in solving physics problems involving variable acceleration.
TPHughes
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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 traveling x, car will have velocity v.


Homework Equations



a = m - nv2 <- given equation

and vt = sqrt(m/n)

x = -0.5ln(1 - (v/vt)2) <- equation need to find

The Attempt at a Solution



I have attempted to substitute in a = d/dx(0.5v2) and then integrate with respect to x but had no luck.

Am stuck with where to go next! Any help would be greatly appreciated!
 
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Welcome to PF!

Hello TPHughes! Welcome to PF! :smile:
TPHughes said:
a = m - nv2 <- given equation

vdv/(m - nv2) = dx :wink:
 
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