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**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.

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.

## Homework 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

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

## 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.