Potential Energy between two protons?

In summary, two protons that are initially 10nm apart and at rest will naturally accelerate away from each other when released. This is due to the conversion of initial potential energy to kinetic energy. The kinetic energy of each proton is equal to 1/2mv^2, and since both protons are moving at the same speed, the total kinetic energy is 2x(1/2)mv^2 = mv^2. Therefore, the potential energy equals the sum of the kinetic energies of the two protons. To calculate the final velocity, the equation v=\sqrt{\frac{kq^2}{md}} is used, with "m" representing the mass of one proton. The correct answer is approximately
  • #1
MCATPhys
58
0
Say you have two protons 10nm apart (at rest). If they are released, they naturally tend to accelerate away from each other. But what's the velocity after they are very far apart?

Basically, all the initial potential energy gets converted to kinetic energy. So I equal them to each other like this:

PE = qV = kq^2/.000000010 = 0.5mv^2

q is the charge of the proton, k is the constant, m is the mass (1.66*10^-27), v is the final velocity

When I solved it, the answer I get is around 5.3*10^3 m/s... but the right answer is supposed to be 3.8*10^3

Can someone please tell me what I am doing wrong... just the theory.
 
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  • #2
When they are very away both protons move with speed v each (this conserves momentum). What is an expression for the kinetic energy of two protons each moving with speed v?
 
  • #3
I tried equaling the initial potential energy to mv^2 (with the mass being that of a proton) - and the answer is still wrong
 
  • #4
Using mv2 for the kinetic energy is the correct way to go. Show what number you got and how and maybe I will be able to find out what went wrong.
 
  • #5
(9*10^9)(1.6*10^-19)^2/.000000010 = (1.66*10^-27)v^2
v = 3725 = which is pretty close to 3.8*10^3 i guess

Is that the answer you are talking about?

But I don't understand why I would use the combined mass of both protons in the kinetic energy 1/2mv^2.

The potential energy should equal the kinetic energy of each of the two protons:
PE = 1/2mv^2 + 1/2mv^2 (m is the mass on of one proton)
PE = mv^2

But why do we use the combined mass for m?
 
  • #6
MCATPhys said:
(9*10^9)(1.6*10^-19)^2/.000000010 = (1.66*10^-27)v^2
v = 3725 = which is pretty close to 3.8*10^3 i guess

Is that the answer you are talking about?
I got 3795 m/s - the difference is probably due to rounding off.

But I don't understand why I would use the combined mass of both protons in the kinetic energy 1/2mv^2.
Anything that moves has kinetic energy. Here we have two protons moving at the same speed. Proton 1 has kinetic energy (1/2)mv2 and proton2 also has kinetic energy (1/2)mv2. Their total kinetic energy is (1/2)mv2 +(1/2)mv2 = 2x(1/2)mv2 = mv2

The potential energy should equal the kinetic energy of each of the two protons:
PE = 1/2mv^2 + 1/2mv^2 (m is the mass on of one proton)
PE = mv^2
You are confused. The potential energy should equal the sum of the kinetic energies of the two protons. That's what
PE = 1/2mv^2 + 1/2mv^2
is saying.
But why do we use the combined mass for m?
That's because the masses are the same and the speeds of the protons are the same because of momentum conservation. If the speeds and masses were not the same, you would have to write

PE = (1/2)m1v12 + (1/2)m2v22.
 
  • #7
I understand what you are saying... the initial potential energy should equal the sum of the kinetic energies of each of the two protons... which means..

PE = 1/2mv^2 + 1/2 mv^2 (but each represents one proton, therefore the m should be for one proton... why do we use the mass of both protons for m?)

btw... you are being very helpful... and thanks so much :)
 
  • #8
nevermind nevermind - bit of a confusion in the book... now i get it... we were using the single mass only
 
  • #9
btw... did you round off to get 3795 m/s - because i didn't do any rounding off
 
  • #10
MCATPhys;2724121... said:
PE = 1/2mv^2 + 1/2 mv^2 (but each represents one proton, therefore the m should be for one proton... why do we use the mass of both protons for m?)
I am not sure what you are asking here. In the equation, the symbol "m" stands for "the mass of one proton" and can be replaced with 1.67x10-27 kg.
 
  • #11
MCATPhys said:
btw... did you round off to get 3795 m/s - because i didn't do any rounding off
I did the calculation on a spreadsheet and got 3794.733192. I calculated

[tex]v=\sqrt{\frac{kq^2}{md}}[/tex]
 
  • #12
kuruman said:
I did the calculation on a spreadsheet and got 3794.733192. I calculated

[tex]v=\sqrt{\frac{kq^2}{md}}[/tex]

yeah - that looks right... but I'm guessing you meant q^2 in the above equation
 
  • #13
haha - u got it
 

Related to Potential Energy between two protons?

1. What is potential energy between two protons?

The potential energy between two protons is the amount of energy that is stored in the system due to the interaction between the two particles. It is a measure of the work that would be required to move the protons away from each other.

2. How is the potential energy between two protons calculated?

The potential energy between two protons is calculated using Coulomb's law, which states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This formula takes into account the electric charge of the protons and the distance between them.

3. Does the potential energy between two protons change with distance?

Yes, the potential energy between two protons changes with distance. As the distance between the protons increases, the potential energy decreases. This is because the force between the protons decreases as the distance increases, resulting in a decrease in potential energy.

4. How does the potential energy between two protons affect their behavior?

The potential energy between two protons affects their behavior by determining their attraction or repulsion towards each other. If the potential energy is positive, the protons will repel each other, and if it is negative, they will attract each other. This potential energy also plays a role in determining the stability of an atom or molecule.

5. Can the potential energy between two protons be converted into other forms of energy?

Yes, the potential energy between two protons can be converted into other forms of energy, such as kinetic energy. When the protons move towards or away from each other, their potential energy is converted into kinetic energy. This conversion of energy is the basis of many physical phenomena, including chemical reactions and nuclear reactions.

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