# Metal block sliding horizontally

## Homework Statement

A metal block of mass m slides on a horizontal surface that has been lubricated
with a heavy oil so that the block suffers a viscous resistance that varies as the 3/2
power of the speed: F(v) = -cv3/2

If the initial speed of the block is vo at x = 0, show that the block cannot travel farther than 2mvo1/2/c

## Homework Equations

F = ma = m$\frac{dv}{dt}$

## The Attempt at a Solution

So, I took

m$\frac{dv}{dt}$ = -cv3/2
I rearranged it and got

dv/v3/2= $\frac{-c}{m}$dt

I've tried integrating this and I can't seem to end up with the right answer. I'm totally lost. Anyone have any ideas?

EDIT: No help at all?

Last edited:

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UltrafastPED
Gold Member
You are not integrating wrt the correct variables; you need dx, not dt.

Use the chain rule to write dv/dt = dv/dx dx/dt = dv/dx v.

Then when you rearrange the terms you will be integrating dv/sqrt(v) = -c/m dx
The integrals run (v0, v_final) and (0, d); if we carry the integral to v_final = 0 then the distance covered will be d_final.

When I do this I get 2m/c sqrt(v0) = d_final, which supports the expected conclusion.

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