# Potential Energy Homework: Particle x Separation and Kinetic Energy E

• Parallel
In summary: It is also worth noting that your expression for Ef is not entirely true. It is only true when the particles are seperated by an infinite distance. You should keep this in mind. Apart from that, you are on the right track.In summary, the potential energy between two particles x and P is given by the equation U(r) = a/r^3 - b/r^2, where a and b are positive constants. The total energy required to separate the particles can be found by taking the limit as r tends to infinity and integrating the potential energy equation. When the particles are at an infinite distance apart, the potential energy is zero. To find the maximum distance between the particles, the total energy is conserved,
Parallel

## Homework Statement

Particle x is bounded to another particle P by force which is dependent on their distance 'r',the potential derived from this force is:

U(r) = a/r^3 - b/r^2

a,b>0

(a)how much energy is required to separate the particles?.
(b)suppose particle x has kinetic energy E,how close/far can he get relative to P?(no need to solve the equation,just write it)

## The Attempt at a Solution

well actually I don't even have a clue on where to start.
for (a) my only thought is that I need to see what happens when r-->infinity.
but I don't know where to plug it,(the force,the potential energy)?

I really need some help

thanks

well your on the right track except its not simply the function as it tends to infinity, as that would be zero, but rather the sum of all of em as it tends to infinity. which calculus function does this for you?

-G

You should take $$\lim_{r\rightarrow \infty} \int \frac{a}{r^{3}} - \frac{b}{r^{2}} \; dr$$

why are you integrating the potential(I don't understand the physical meaning of it)?

any hints for (b)?

thanks!

Last edited:
EDIT - the original version of this was garbage. Apologies!

From the definition of potential U, the force on the particle at any point = -dU/dr

Work = force times distance.

Work done in moving from radius r1 to r2 = integral (force.dr) = U(r1) - U(r2).

In both parts of the question, I think you need to know the initial distance between the particles.

Last edited:
maybe I'm wrong,but if you integrate the potential you don't get the force!
if you differentiate the potential you get the force.

Yeah, sorry for complete brain failure. See my edited post #5.

Let us try an take a less mathematical approach. By definition, what is the potential energy of the particles when they are at an infinite distance apart (i.e. when $r\to\infty$)?

Last edited:
well,the potential energy is zero.

as you get them closer(i.e doing work),you increase their potential energy.

AlephZero:
the intial distance is not given in the problem

Parallel said:
well,the potential energy is zero.

as you get them closer(i.e doing work),you increase their potential energy.
Correct, so the work you do on the particle would be the change in potential energy, i.e;

$$W = U_{f}-U_{i} = U_{r\to\infty} - U_{r'}$$

Make sense?

The problem text should contain the fact that the initial distance between the particle is r' .

Hootenanny said:
Correct, so the work you do on the particle would be the change in potential energy, i.e;

$$W = U_{f}-U_{i} = U_{r\to\infty} - U_{r'}$$

Make sense?

yea it's clear now thanks :)

any suggestions for (b) ?

Hint: the total energy is conserved, see how you could use that.

o.k so the energy is conserved,let's assume that the distance between the particles is R.so becuase energy is conserved:

E + a/R^3 - b/R^2 = a/r^3 - b/r^2
left side is 'Ei',and the right side is Ef(no kinteic energy just potential energy)

is this o.k?

thanks for your help guys

I would agree with that in general. However, I would make one small change;you should label your r on the RHS something different, such as r'' since r is a variable in your equation.

## 1. What is potential energy?

Potential energy is the energy that a system possesses due to its position or configuration. It is stored energy that can be converted into other forms, such as kinetic energy, as the system changes.

## 2. How is potential energy related to particle separation?

The potential energy between two particles is directly proportional to the distance between them. As the particles are brought closer together, the potential energy increases. As they are separated, the potential energy decreases.

## 3. What is the equation for calculating potential energy?

The equation for calculating potential energy is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height or distance of the object from a reference point.

## 4. How does potential energy affect kinetic energy?

Potential energy can be converted into kinetic energy. As an object falls, its potential energy decreases while its kinetic energy increases. This is because the potential energy is being converted into the energy of motion.

## 5. What is the relationship between potential energy and work?

Work is the transfer of energy. The work done on an object is equal to the change in its potential energy. As potential energy increases, work is done on the object, and as potential energy decreases, work is done by the object.

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