# Car accelerated by repulsion of two point charges

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1. Oct 13, 2015

### wackyvorlon

At the outset, I want to explain that this is a problem I came up with myself. It's not actually homework, and I suspect it is deeply conceptually flawed in some manner that I have yet to determine.

1. Two point charges of like polarity, $q_1 = q_2 = 1C$, start out separated by distance $x = 1m$. $q_2$ is attached to a car of mass $m=1000kg$. When released, the car is accelerated by the force repelling the two charges. Find a function $v(t)$ which gives the speed at time t.

2. Relevant equations

$$F = k \frac{q_1 q_2}{x^2}$$

$$a = \frac{F}{m}$$

$$v = a t$$

Potential Energy

$$U = k \frac{q_1 q_2}{x}$$

Kinetic Energy

$$K = \frac{1}{2} m v^2$$

3. The attempt at a solution

This has been giving me fits. I keep ending up in circular definitions. Firstly, I approach from the perspective of energy, $U_0 = K_f$. The end result of that was that the final speed should be $3.16*10^3 \frac{m}{s}$.

$F$ becomes: $$F = \frac{k}{x^2}$$

Inserting into Newton's second law I get:

$$a = \frac{k}{m x^2}$$

Then:

$$v = \frac{k}{m x^2} t$$

You'll notice my problem. Through some means, I have to express $x$ in terms of $t$, but every idea I've had relies, ultimately, on $x$. Truthfully, to list the approaches I've tried in detail here would require quite some typing. I feel intuitively that there ought to be some way to solve this, but frankly I'm at a loss. Any assistance you can provide in pointing me in the right direction would be greatly appreciated.

2. Oct 13, 2015

### haruspex

Ok.
Not ok.
x is a variable. You cannot integrate x-2 by simply multiplying by t.

You can use energy conservation to find the velocity as a function of position, but getting it as a function of time is quite tricky.

3. Oct 15, 2015

### wackyvorlon

Thanks! Your help is greatly appreciated.

I've been wondering if perhaps this would be a good application of the laplace transform?

4. Oct 15, 2015