# Find the antiderrivative of $\int \frac{v^2-v_o^2}{2\frac{K_e\frac{q_1q_2}{r^2}}{m} } dr$

I'll admit I'm not the best when it comes to calculus.  Anyway, I'm trying to model how close two nuclei will get to each other for a given starting velocity.  Since they're charged particles, the force between them abides by Coloumb's law: $f =K_e\frac{q_1q_2}{r^2}$.  Since nuclei are not quantum particles, they also abide by Newton's second law: $f=ma$.  Therefore:

$ma=K_e\frac{q_1q_2}{r^2}$

Solving for a:

$a =\frac{K_e\frac{q_1q_2}{r^2} }{m}$

We know from one of the kinematic equations:

$v^2=v_o^2+2a\triangle x$

Transposing to solve for $\triangle x$:

$\triangle x=\frac{v^2-v_o^2}{2a}$

Plugging in the equation for acceleration, we get:

$\triangle x=\frac{v^2-v_o^2}{2\frac{K_e\frac{q_1q_2}{r^2}}{m} }$

Since $\triangle x$ is dependent on acceleration, which is constantly increasing as distance is decreasing, if I remember calculus correctly, I can take the definite integral with respect to r to get the total distance each nuceli travels:

$\int_{a}^{b} \frac{v^2-v_o^2}{2\frac{K_e\frac{q_1q_2}{r^2}}{m} } dr$

To get that, I'd need to take the antiderrivative:

$\int \frac{v^2-v_o^2}{2\frac{K_e\frac{q_1q_2}{r^2}}{m} } dr$

The problem is, that's the extent of my mathematical ability.  We never really got to problems this complex in the sole calculus class I've taken.

So, I have two questions:
1. Since every variable except r is an independent variable, can they simply be treated as constants, and
2. What is the antiderrivative?

• Do you know the mass of both nuclei? If so, I think you can do much better by just using conservation of energy. The equations of motion get a lot more complicated than what you described.

• Your approach does not work because the velocity depends on the distance, you cannot just treat it as a constant.

• I don't think there is enough information to solve this problem. Does one nucleus just stay still the whole time, while the other moves towards it, or are both of them free to move? Does one start still while the other moves, or do both of them have non-zero initial velocities?

• I solved the problem in a couple simple cases, let me know if that is enough or you have different specifications.