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## Homework Statement

The proposed design for an energy- absorbing bumper for a car exerts a decelerating force of magnitude

*bs + cv*on the car when it collides with a rigid obstacle, where

*s*is the distance the car travels from the point where it contacts the obstacle and

*v*is the car's velocity. Thus the force exerted on the car by the bumper is a function of the car's position and velocity.

(a) Suppose that at

*t = 0*the car contacts the obstacle with initial velocity u. Prove that the car's position is given as a function of time by

*s*(equation provided below). To do this, first show that this equation satisfies Newton's second law. Then confirm that it satisfies the initial conditions

*s = 0*and

*v = u*at

*t = 0*

(there's also (b) but maybe I could do it by myself if this is solved)

## Homework Equations

*F = bs + cv*

[tex] s = \frac{u}{2h}[e^{-(d-h)t} - e^{-(d+h)t}] [/tex]

[tex] d = \frac{c}{2m} [/tex]

[tex] h = \sqrt{d^2 - \frac{b}{m}} [/tex]

b and c are constants

## The Attempt at a Solution

Well I first tried substituting with s in bs + cv to try and simplify it to ma then I realized there wouldn't be an a in the left hand side of the equation, so I substituted a with dv/dt and I integrated both sides, but I was still unable to simplify it (because of the

*e*s!) - (Edit: I obviously couldn't have integrated correctly because I can't separate the variables). Is this even the way to go about it? I tried starting with bs + cv = ma and again substituting with a = v dv/ds but I was unable to separate the variables. Please help? It's important.

I also tried differentiating s with respect to time and then substituting v in bs + cv with the result, but I still can't simplify.

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