# Deriving equations of motion from power and mass

1. Jul 6, 2013

### Jewber

I'm terrible at calculus and am trying an exercise to hopefully help me understand it better. I want to derive the equations of acceleration, velocity and position of a car with known power and mass. As the car's speed increases, the acceleration will decrease.

force = mass/acceleration
power = force*velocity

So acceleration = power/(velocity*mass)
velocity = ?
position = ?

The integral of acceleration is the velocity but how is the integral done in this case since velocity is an unknown function?

2. Jul 6, 2013

### HallsofIvy

Staff Emeritus
Assuming you are talking about a fixed power and mass, we can write "acceleration= power/(velocity*mass)" as $dv/dt= P/(vm)$ and separate- $dv/v= (P/m)dt$. Integrating both sides, $ln(v)= (P/m)t+ c$ or $v= Ce^{(P/m)t}$ where $C= e^c$ is a constant equal to the initial velocity. Integrating that with respect to t, $x= C(m/P)e^{(P/m)t}+ D$.

3. Jul 7, 2013

### Jewber

If initial velocity is zero at time = 0, what is the constant C?

ln(v)=(P/m)t+c
ln(0)=(P/m)(0)+c
c=ln(0)?