Speed at which two charges collide

[djrkeys4]

Homework Statement

Point charge A of mass m for which q=-Q is held in place as point charge B of mass m for which q=+Q is released from rest at a distance x. What is the speed of charge B as it collides with charge a? (This isn't an actual problem we have, just something that I was wondering as we're starting E&M)

Homework Equations

Fe=kqq/r^2

E=kqq/r

Potential energy = Kinetic energy

The Attempt at a Solution

I started out with an a(x) equation but didn't get beyond that as I couldn't find how to work with functions of time instead of position. Can you do this by just converting the potential energy at rest to kinetic energy? I feel like there should be more to it because the acceleration goes to infinity as b gets closer and closer to a.

 

[djrkeys4]

Just gave it some more thought and realized the U -> KE idea doesn't make sense (U increases as x decreases)... can anyone point me in the right direction?

 

[djrkeys4]

Alright, new idea: would it be easier instead to find the time it takes for a positive charge to collide with a negative charge that is held in place a distance r away from where the positive charge starts?

It's probably easier to find the time it takes the moving charge to move from r to r-x away from the stationary charge, so to find it could we take the limit as x approaches r of some equation? I have no idea what that would be, though.

 

[SammyS]

Homework Statement

Point charge A of mass m for which q=-Q is held in place as point charge B of mass m for which q=+Q is released from rest at a distance x. What is the speed of charge B as it collides with charge a? (This isn't an actual problem we have, just something that I was wondering as we're starting E&M)

Homework Equations

Fe=kqq/r^2

E=kqq/r

Potential energy = Kinetic energy

The Attempt at a Solution

I started out with an a(x) equation but didn't get beyond that as I couldn't find how to work with functions of time instead of position. Can you do this by just converting the potential energy at rest to kinetic energy? I feel like there should be more to it because the acceleration goes to infinity as b gets closer and closer to a.

Hello djrkeys4. Welcome to PF !

I doubt that you will ever see this problem in a textbook. The potential energy → -∞ as x → 0 . Therefore, the kinetic energy → +∞ as x → 0 .

Alright, new idea: would it be easier instead to find the time it takes for a positive charge to collide with a negative charge that is held in place a distance r away from where the positive charge starts?

It's probably easier to find the time it takes the moving charge to move from r to r-x away from the stationary charge, so to find it could we take the limit as x approaches r of some equation? I have no idea what that would be, though.

I haven't worked it out, but I'm pretty sure that it is possible to find the time it takes for the particles to collide.

 

[asteriatic]

To determine the speed at which two charges collide, we can use the principle of conservation of energy. At the moment of collision, the potential energy of the system is converted into kinetic energy. We can set up the equation as follows:

Potential energy at rest = Kinetic energy at collision

This can be written as:

kqQ/x = (1/2)mv^2

Where k is the Coulomb's constant, q and Q are the charges of the two particles, x is the distance between them, m is the mass of the particles, and v is the speed at collision.

Solving for v, we get:

v = √(2kqQ/mx)

Therefore, the speed of charge B at collision with charge A will depend on the masses and charges of the particles, as well as the distance between them. As the distance between the particles decreases, the speed at collision will increase, approaching infinity as the distance approaches zero. This is due to the inverse square relationship of the Coulomb force, which becomes increasingly stronger as the distance decreases.