A rock with mass m slides with initial velocity v0 on a horizontal surface. A retarding force F that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock (F = -kv1/2) . 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.
Integrate both sides and solve for the constant C
V= (k2 t2 )/ (4m2 ) - (ktv01/2)/m + v0
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.