Retarding Force and Velocity of a Sliding Rock

[carlee172]

Homework Statement

A rock with mass m slides with initial velocity v_o on a horizontal surface. A retarding force F_R that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock F_R=-kv^1/2.
A) Find expression for the velocity of the rock as a function of time.
Express your answer in terms of the variables m, v_o, k, and t.

Homework Equations

F=ma

v=v_o +at

etc..

The Attempt at a Solution

vbasically included solving and substituting F=ma, F_R=-kv^1/2 and v=v_o +at
in the end i got v=v_o -kv^1/2t/m

obviouslt this isn't right because i have v on both sides of the equation.. help

 

[Redbelly98]

Welcome to PF.

v=v_o +at doesn't work here, because you need a constant acceleration for that equation.

F=ma is a good start. You'll also need to use some calculus to solve this one.

 

[carlee172]

ok, so:

-k$$\sqrt{}v$$=m(dv/dt)
then integrate the first side from 0 to t and the left side from v_o to v
and then solve for v, i got:
v=((-kt-2(v^1/2)m)/2m)²

 

[carlee172]

ok so i got the answer v=(4m²v₀-4ktmv^1/2+k²t²)/4m²
now, i have to find the integral (distance). for t, the rest are constants i guess

 

[Redbelly98]

One problem with that expression is that it's supposed to be an expression for v in terms of t and the other variables, i.e. given t, m, k, and v0, you should be able to calculate v.

But you have v there on the right-hand-side, so it's not really an equation for v.

ok, so:

-k$$\sqrt{}v$$=m(dv/dt)

Let's go back to this equation. I didn't really follow what you did after that, but I would separate the two variables, v and t, before integrating. I.e., put all the v's (including the dv term) on one side of the equation, and put dt on the other side. Then integrate.