Kinetics Problem: Non-constant force (calculus)

[Lola Luck]

Homework Statement

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 = -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= (k² t² )/ (4m² ) - (ktv₀^1/2)/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.

 

[haruspex]

V= (k² t² )/ (4m² ) - (ktv₀^1/2)/m + v₀
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.

Not at all messy, it turns out.

Is there another way?

Slightly. After integrating dv/(v^1/2)=(-kdt)/(m), and determining the constant, substitute v = 0 without converting it to the quadratic form.

 

[Lola Luck]

I made a calculation error and didn't realize until now that solving v(t) with a quadratic actually comes out neatly.

 

[Lola Luck]

Thanks for your help

 

[haruspex]

Thanks for your help

My pleasure.