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!