# Solve Bouncing Electrons: Find r_min with q, m, v, and k

• fubag
In summary, the problem deals with two electrons with mass m and charge q, initially moving towards each other with nonzero speed, v, along the x-axis. As they approach each other, their electric repulsion causes them to slow down. The minimum separation they reach, r_min, can be determined by using the conservation of mechanical energy and momentum. By setting the initial potential energy, U1, to zero and solving for r_min, it can be expressed as (kq^2)/(4mv^2), where k is the electric constant.
fubag
[SOLVED] Bouncing Electrongs

Two electrons, each with mass m and charge q, are released from positions very far from each other. With respect to a certain reference frame, electron A has initial nonzero speed v toward electron B in the positive x direction, and electron B has initial speed 3v toward electron A in the negative x direction. The electrons move directly toward each other along the x-axis (very hard to do with real electrons). As the electrons approach each other, they slow due to their electric repulsion. This repulsion eventually pushes them away from each other.

What is the minimum separation (r_min) that the electrons reach?

All I understand so far is that both electrons are moving at the same nonzero speed in the same direction.

I need help starting this question. I understand that U1 + K1 = U2 + K2 for conservation of energy.

I also understand that U = kqQ/r for a point charge. So in this case I believe U1 = -ke^2/r, and then K1 = mv^2/r. Also I believe U2 = -ke^2/r and K2 = -3mv^2/r.

Please tell me if these are correct assumptions, and what I need to do next.

Thanks!

fubag said:
All I understand so far is that both electrons are moving at the same nonzero speed in the same direction.
If you mean at the moment of minimum separation: good! You'll have to find that speed.

I need help starting this question. I understand that U1 + K1 = U2 + K2 for conservation of energy.
Good.

I also understand that U = kqQ/r for a point charge. So in this case I believe U1 = -ke^2/r, and then K1 = mv^2/r. Also I believe U2 = -ke^2/r and K2 = -3mv^2/r.
Two problems: (1) the potential energy should be positive, since the charges are the same; (2) KE is always positive.

Please tell me if these are correct assumptions, and what I need to do next.
Hint: What else is conserved?

ok error on the U1 should be ke^2/r, and K1 = mv^2 and both of these added together must equal U2 = ke^2/r added with K2= 3mv^2/r, where we can take r as minimum.

So since these must be conserved I could just solve for r_min?

fubag said:
ok error on the U1 should be ke^2/r, and K1 = mv^2 and both of these added together must equal U2 = ke^2/r added with K2= 3mv^2/r, where we can take r as minimum.
KE = 1/2 mv^2, not mv^2.

So since these must be conserved I could just solve for r_min?
Not until you determine the common speed of the electrons when they reach r_min. Hint: What else is conserved?

I am not sure what else must be conserved...given the potential and kinetic energy, I can only think of charge and mass remaining conserved.

Nope, there's one more conservation law that you're forgetting.

conservation of linear momentum?

Absolutely!

ok so kinetic energy and momentum must be conserved in this system.

so (1/2)mv^2 + (3/2)mv^2 = mv^2 (since both electrons move at the same speed in the same direction when they are at their minimum separation)?

change in momentum final must be equal to change in momentum initial.

i think i am doing something wrong

fubag said:
ok so kinetic energy and momentum must be conserved in this system.
Total mechanical energy is conserved, not kinetic energy.

Use conservation of momentum to figure out the speed the electrons must have at minimum separation. Then you can use conservation of energy to figure out the separation.

ok this is what I got so far:

m1v1 + m2v2 = m1v1_f + m2v2_f

based on the information given in the problem:

m1=m2, v1 = v, v2 = 3v, and at their minimum separation they are going the same speed in the same direction so...

mv + 3mv = mv + mv
4mv = 2mv; this equality doesn't make much sense to me, i believe i am still making a mistake somewhere

direction counts!

fubag said:
m1=m2, v1 = v, v2 = 3v,
They are moving towards each other--opposite directions; thus they should have opposite signs.

ok so redoing it once again...

mv - 3mv = 2mv_2

v_2 = -v, speed at minimum separation?

Sounds good.

ok now that I have the speed during minimum separation, i use the conservation of mechanical energy laws...

as written before

U1 + K1 = U2 + K2; K1 = (mv^2/2) + (m(-3v)^2/2); K2 = (mv^2/2) + (mv^2/2)

(ke^2/r) + (10mv^2/2) = (ke^2/r) + (mv^2)

but in my equation above, I will not be able to solve for r...so once again I think I am missing something

Since the electrons start from very far away, you can say U1 = 0 (r_1 = infinity). Then just solve for r_2 (which is r_min) in terms of k,q,m, & v.

ok thank you so much for your help so far Doc Al

this is what I know come up with:

U1 + K1 = U2 = K2

U1 = 0

K1 = 5mv^2; using kinetic energy of both electrons

U2= kq^2/r

K2 = mv^2

Solving for r_min I get (kq^2)/(4mv^2). Does this make sense?

fubag said:
K2 = mv^2
How did you get this? What's the speed of each electron at r_min?

You're almost done.

I got K2 by taking -v as the speed of the electron at r_min; and since there are two electrons I added their kinetic energy together. Is this ok to do? so (mv^2/2) + (mv^2/2) = mv^2?

You're right. (You caught me sleeping!)

## What is the equation for solving bouncing electrons?

The equation for solving bouncing electrons is r_min = (q^2*k)/(4*pi*m*v^2), where r_min is the minimum distance between the electron and the nucleus, q is the charge of the electron, m is the mass of the electron, v is the velocity of the electron, and k is the Coulomb's constant.

## What is the significance of finding r_min?

Finding r_min allows us to determine the closest possible distance between the electron and the nucleus during its bouncing motion. This can help us understand the behavior of electrons in different environments and can be used in various applications such as designing electronic devices.

## How do the variables in the equation affect the value of r_min?

The value of r_min is directly proportional to the charge of the electron (q) and the Coulomb's constant (k), and inversely proportional to the mass of the electron (m) and the square of the velocity (v^2). This means that any changes in these variables will affect the value of r_min.

## What are the units of measurement for each variable in the equation?

The units of measurement for r_min are meters (m), q is measured in coulombs (C), m is measured in kilograms (kg), v is measured in meters per second (m/s), and k is measured in newton-meter squared per coulomb squared (Nm^2/C^2).

## Can this equation be used for other particles besides electrons?

Yes, this equation can be used for any charged particle as long as the values of the variables are known. However, the mass (m) in the equation will need to be replaced with the mass of the specific particle being studied.

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