This is not a homework question, it is from a past paper which I am using to practice. The question is shown in the image below:


I really don't know much about mechanics, so I don't even know where to start with this one. Any help much appreciated.


  • $\begingroup$ Do you need help deriving the differential equation (which is more of a physics question) or solving the differential equation (which is more of a math question)? $\endgroup$ – user137731 May 17 '15 at 17:01
  • $\begingroup$ Mainly struggling to derive the DE $\endgroup$ – M Smith May 17 '15 at 17:02

Well the particle will always be subject to gravity, so you've got the $-g$ term there, since we're 'pointing upwards'.

Then we're told the particle experiences a force of magnitude $mkv$ resistive to its upward motion. From Newton's second law with constant mass $F=m\frac{dv}{dt} \Rightarrow \frac{dv}{dt} = \frac{F}{m}$ so we get the second term $-kv$.

So we've done the first bit.

Now remember that $v = \frac{dx}{dt}$ which means that to get $x$ you're going to need to integrate $\frac{dv}{dt}$ twice with respect to $t$. We have the initial condition $v(0)=v_0$ too.

The differential equation you need to solve is therefore

$$ \frac{d^2 x}{dt}=-g-k\frac{dx}{dt} $$

  • $\begingroup$ Okay, that really helps. Thanks. $\endgroup$ – M Smith May 17 '15 at 17:12
  • $\begingroup$ No problem bud. $\endgroup$ – jonbaldie May 17 '15 at 18:24

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