# Potential Energy of a Particle

1. Oct 29, 2007

### Yuravian

1. The problem statement, all variables and given/known data
A 0.40-kg particle moves under the influence of a single conservative force. At point A where the particle has a speed of 10 m/s, the potential energy associated with the conservative force is +40 J. As the particle moves from A to B, the force does +25 J of work on the particle. What is the value of the potential energy at point B?

The problem, for me is that there is no diagram included on the homework sent out in the email from the professor and I think the proper problem would have had a diagram of the path. I may be wrong though.

2. Relevant equations
Well, this is clearly a work and kinetic energy problem, so:
$$W=\Delta U+\Delta K$$
$$W=Fdcos\theta$$ (not sure if it applies here)
$$\Delta K=1/2mv^{2}_{o}-1/2mv^{2}_{f}$$
$$\Delta U=mgh_{o}-mgh_{f}$$ (again, I don't think this applies)
3. The attempt at a solution
straighten out my variables:
$$v_o=10 m/s$$
$$K_o=1/2mv^2_o=20 J$$
$$U_o=40 J$$
$$W=25 J=U_f-U_o+K_f-K_o$$

So at this point, I do not have the acceleration of the particle (cannot find F or v final), and no information on the final state of the particle other than the amount of work done on it. Where should I go from here?

2. Oct 29, 2007

### Feldoh

Actually I don't think you need a diagram to solve this. Think about: if we assume that the energy is conserved in the system then the total energy is equal to the kinetic + potential energy. We know the potential energy at point a and can calculate the kinetic energy at point a as well...

It might be helpful to know that

$$W = \Delta K$$

Last edited: Oct 29, 2007
3. Oct 29, 2007

### qspeechc

Sorry, I just cut that out. I mis-read the question. I still do not think it can be solved though.

4. Oct 29, 2007

### Feldoh

No, I really think it can be solved. You can find the total mechanical energy, and then find the kinetic energy at point b and from that find the potential at point b.

5. Oct 29, 2007

### qspeechc

How would you find the kinetic energy at B?

6. Oct 29, 2007

### Feldoh

At point A: $$KE = \frac{1}{2}mv^2 = \frac{1}{2}(0.40kg)(10\frac{m}{s})^2 = 20J$$

$$E_{mec} = KE + U = 20J + 40J = 60J$$

We know the work done from point A to B and
$$W = \Delta K = \frac{1}{2}m(v_f^2-v_o^2)$$

We need to solve for the velocity of the particle at point B... we get:

$$v_{f}^2 = \sqrt{\frac{2W}{m}+v_o^2} = \sqrt{\frac{2(25J)}{0.40kg}+10\frac{m}{s}^2} = 15\frac{m}{s}$$

So KE at B: $$\frac{1}{2}mv^2 = \frac{1}{2}(0.40kg)(15\frac{m}{s})^2 = 45J$$

Then find the potential energy

$$U = E_{mec} - KE = 60J - 45J = 15J$$

7. Oct 29, 2007

### qspeechc

As Yuravian states, work is change in mechanical energy, that is
W = ∆KE + ∆U

Not:
W = ∆KE
as you have stated.

8. Oct 29, 2007

### Feldoh

We're talking about a single conservative force acting on the particle. Conservative forces wont change the total amount of mechanical energy we have.

∆KE + ∆U = ∆E = W (by non conservative forces)

Last edited: Oct 29, 2007
9. Oct 29, 2007

### qspeechc

So? Can a single conservative force not change the potential energy? What if I lift a bag of sand vertically in a vacuum?

10. Oct 29, 2007

### Feldoh

It will change the potential energy, but it will not change the total amount of energy in the system.

Last edited: Oct 29, 2007
11. Oct 29, 2007

### qspeechc

If I do work in lifting the bag of sand, initially at rest on the floor, can I not lift it vertically some distance AND give it some speed, so that now it has BOTH potential AND kinetic energy, where originally it had none?

12. Oct 29, 2007

### Feldoh

Sure, but ∆KE = -∆U = W, U is going to be changing negatively since the work is in the opposite direction of the force of gravity. The total energy will still be 0.

Last edited: Oct 29, 2007
13. Oct 29, 2007

### Yuravian

Ah, I see... I missed the part about the force being conservative. If a force is conservative, then W = ∆KE = -∆U, since E=KE+U and E doesn't change at all, right? so would it not be easier to simply say $$U_f = U_o+\Delta U = 40J - 25J = 15 J$$?

Last edited: Oct 29, 2007
14. Oct 29, 2007

### Yuravian

Wait a minute. If your v is high and stays constant, wouldn't your U increase while K stayed constant? Correct me if I'm wrong, but if you lift a bag of sand, the force you put on it isn't conservative in the first place.

15. Oct 29, 2007

### Feldoh

Oh I think he meant like throwing it upwards. I think you're right though the force of the push is not conservative...

Haha, yeah your way sure beats mine :P