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## Homework Statement

A rock with mass

*m*slides with initial velocity

*v*on a horizontal surface. A retarding force

_{0}*F*that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock (F = -kv

^{1/2}) . a) Find expression for the velocity of the rock as a function of time. b) Find expression for the position of the rock as a function of time. c) In terms of

*m,k,*and

*V0*, at what time will the rock come to rest? d) In terms of m ,k and V0, what is the distance of the rock from its starting point when it comes to rest?

**2. Relevant equation**

f=ma A= dv/dt

## The Attempt at a Solution

I solved parts a and b and I'm fairly certain my answers are correct.

F=−kv^(1/2)

a=dv/dt= (-kv^(1/2))/m

dv/(v^1/2)=(-kdt)/(m)

Integrate both sides and solve for the constant C

V= (k

^{2}t

^{2})/ (4m

^{2}) - (ktv

_{0}

^{1/2})/m + v

_{0}

Then I integrated again to find position as a function of time.

The only way I can think to solve parts c and d is to solve V(t)=0 but this would require the quadratic equation and be very messy. Then, to solve part d, I would have to plug in answer to c into x(t) which is even messier. Is there another way? Am I missing something?

Any help would be great. Thank you.

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