# Finding the maximum acceleration

## The Attempt at a Solution

I don't know how to make the equations here.

When the balls are penetrating each other, a cavity is being formed due to overlap of the negative and positive charges. The size of this cavity keeps on increasing. The trouble is I cannot figure out the instant of maximum velocity and the equations to write down. I need a few hints to begin with.

Any help is appreciated. Thanks!

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Does it work now?

If a function reaches a maximum, what happens to its derivative? What does that mean in this case?

If a function reaches a maximum, what happens to its derivative? What does that mean in this case?
The derivative is zero but to find the maximum, I need the function. Can you please give me a few hints about finding the function? :)

What is the derivative of velocity? What does that mean, physically, its being zero? When/where can that happen in this situation?

What is the derivative of velocity?
Acceleration.

What does that mean, physically, its being zero?
It means acceleration is zero, the body is at rest or is moving with a constant velocity.
When/where can that happen in this situation?
I am not sure so I make a guess. It can happen when both the balls completely overlap. I think when the balls completely overlap, the net charge is zero so no electric force is acting, right?

I agree with that. That gives you some important knowledge: you know at what separation between the centers their speed was 10 m/s. Can you proceed further?

I agree with that. That gives you some important knowledge: you know at what separation between the centers their speed was 10 m/s. Can you proceed further?
The velocity is maximum when the separation is zero, right?

I can't think of anything about the next step. :(

What is the potential energy of the system when the separation is zero?

What is the potential energy of the system when the separation is zero?
Umm...zero?

Why do you think it is zero?

Why do you think it is zero?
There is no charge when separation is zero which led me to think that it is zero. Am I wrong?

No charge means the force is zero, but that does not mean the potential energy is zero. We typically define potential energy in such a way that it is zero at the infinite separation. If it is defined like that, then your suggestion that the potential energy is zero at the zero separation is problematic with respect to conservation of energy. Can you see that?

No charge means the force is zero, but that does not mean the potential energy is zero. We typically define potential energy in such a way that it is zero at the infinite separation. If it is defined like that, then your suggestion that the potential energy is zero at the zero separation is problematic with respect to conservation of energy. Can you see that?
I am not sure so I thought of starting with the definitions.

$$U(R)-U(0)=-\int_0^R \frac{kQq}{r^2}dr$$
(where both Q and q are positive)

$$U(R)-U(0)=\lim_{r\rightarrow 0}\frac{kQq}{r}-\frac{kQq}{R}$$

I don't see how defining U(0)=0 leads to complications. :(

I am not sure so I thought of starting with the definitions.

$$U(R)-U(0)=-\int_0^R \frac{kQq}{r^2}dr$$
(where both Q and q are positive)
I am not sure what that equation means. What is ##R##? Is it the size of the sphere? Then I do not see where you factor in the separation. If it is the separation, then I do not see how you account for separations less than the diameter of the spheres.

I don't see how defining U(0)=0 leads to complications. :(
Well, if it is zero both at the infinity and at the zero separation, then the kinetic energy must also be zero at the zero separation if it was zero at the infinity. Which, I think, quite obviously is an impossible result.

I am not sure what that equation means. What is ##R##? Is it the size of the sphere? Then I do not see where you factor in the separation. If it is the separation, then I do not see how you account for separations less than the diameter of the spheres.
Sorry, I should have been clear, I was taking a general case when a point charge q is taken from r=0 to r=R from a fixed point charge Q.

Well, if it is zero both at the infinity and at the zero separation, then the kinetic energy must also be zero at the zero separation if it was zero at the infinity. Which, I think, quite obviously is an impossible result.
I think I understand your point but I am still clueless about the given problem.

Here is a question: what does the graph of the potential energy vs the separation look like?

Here is a question: what does the graph of the potential energy vs the separation look like?
When the separation between the centre of balls is 2R, the potential energy is ##-kQ^2/(2R)##. For separation greater than 2R, the graph increases and tends to zero at infinity. For separation less than 2R, I am not sure how the graph would look like as they begin to penetrate each other. Would the potential energy still be ##-kQ^2/r## where r is the separation between the centre of balls and r<2R?

Remember that you have already established that the kinetic energy has a maximum at the zero separation. What about the potential energy?

Remember that you have already established that the kinetic energy has a maximum at the zero separation. What about the potential energy?
If potential energy is still given ##-kQ^2/r##, then at zero separation, it shoots to ##-\infty##.

But initially, the energy of system is zero, so kinetic energy at zero separation must be equal to the potential energy.

What is important is that the potential energy is some constant minus the potential energy. So when one is at a maximum, the other is at a minimum. Hence, the potential energy must be at a minimum at the zero separation. It can be also seen from the fact that at the zero separation the force is zero, and the force is the derivative of the potential energy.

Yet another way would be by saying "it is obvious that a stable equilibrium configuration of two equal shape, equal mass, opposite charge clouds is when they coincide completely".

So, what does the total graph look like? Where is it convex, where is it concave? Where is the inflection point? What is its significance?

So, what does the total graph look like? Where is it convex, where is it concave? Where is the inflection point? What is its significance?
I don't understand why you ask me about the graph when it is simply a -1/r relationship.

http://www.wolframalpha.com/input/?i=y=-1/x

The shape in the lower-right quadrant is correct only when the separation is greater than the diameter. At smaller separations, the graph is different. Which is obvious from the fact that at the zero separation it has a minimum, not a singularity.

The shape in the lower-right quadrant is correct only when the separation is greater than the diameter. At smaller separations, the graph is different. Which is obvious from the fact that at the zero separation it has a minimum, not a singularity.
How do I find the expression for potential energy when the balls penetrate each other?

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