An object with mass $m$ receives 11 impulses of strength $p$ at 1 second intervals at $t=0,1,2,\ldots,10$. The differential equation describing the motion of this object is $$m\frac{dv}{dt} = p\sum_{k=0}^{10}\delta(t-k).$$ If the object is initially at rest, find its velocity at time $t\geq 0$.
I can't figure out how to do this.
I'd appreciate it if you could put some effort into this and show any attempts that you've made. I don't want it to seem like I'm doing your homework for you; also, I'm sure more people would be willing to help if you showed some sort of attempt. :-)
Anyways, since $\mathcal{L}\{\delta(t-k)\} = e^{-ks}$, it follows that after taking Laplace transforms of both sides, you're left with
$$m(sV(s)-v(0)) = p\sum_{k=0}^{10}e^{-ks}\implies V(s) = \frac{p}{m} \sum_{k=0}^{10} \frac{e^{-ks}}{s}$$
Can you compute the inverse Laplace transform and take things from here?