wackyvorlon
- 3
- 0
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.
$$ F = k \frac{q_1 q_2}{x^2} $$
$$ a = \frac{F}{m} $$
$$ v = a t $$
Potential Energy
$$ U = k \frac{q_1 q_2}{x} $$
Kinetic Energy
$$ K = \frac{1}{2} m v^2 $$
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.
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.