# Force and energy proofs

1. Sep 6, 2006

### Reshma

Show that for a single particle with constant mass the equation of motion implies the following differential equation for the kinetic energy:
$${dT\over dt} = \vec F \cdot \vec v$$

while if the mass varies with time the corresponding equation is
$${d(mT)\over dt} = \vec F \cdot \vec p$$

Proof:

I was able to prove the first part:
$$T = {1\over 2}mv^2$$
$${dT\over dt} = {1\over 2} {d(mv^2)\over dt} = \vec v \cdot m{d\vec v \over dt} = \vec F \cdot \vec v$$

2. Sep 6, 2006

### tim_lou

what does mT equal in terms of m and v? in terms of p?
mv=p
F=dp/dt

try to play with the equation a little and substitute p in.