Calculating Speed of a Block Pushed Across a Horizontal Surface

[crimpedupcan]

Homework Statement

A block of mass m is at rest at the origin at t=0. It is pushed with constant force F_0 from x=0 to x=L across a horizontal surface whose coefficient of kinetic friction is \mu_k=\mu_0(1-x/L). That is, the coefficient of friction decreases from \mu_0 at x=0 to zero at x=L.
Find an expression for the block's speed as it reaches position L.

The Attempt at a Solution

I ended up with F_{net}=F_0 - mg\mu_0 + \frac{mgx}{L}
I can use this to find a in terms of x, but I don't know what use that would be.

 

[BruceW]

I ended up with F_{net}=F_0 - mg\mu_0 + \frac{mgx}{L}
I can use this to find a in terms of x, but I don't know what use that would be.

That is almost correct. The last term is not quite right. Try checking it, I think you just made a slight mistake. Also, about what to do next - you have an equation for the force on the block, so how could you find the change in kinetic energy of the block?

 

[tia89]

Try writing
$$ a(x)=\frac{d^2 x}{dt^2} $$
and solve the differential equation for $x(t)$... then final speed is
$$ \frac{dx}{dt}|_{x=L} $$

 

[crimpedupcan]

Okay, I've solved the problem using the work/kinetic energy method implied by BruceW. Thanks!
If you're curious I got v=\sqrt{\frac{2F_0L}{m}-Lg\mu_0}

 

[BruceW]

yep. nice work!