Velocity From Electric Potential

[PurelyPhysical]

Homework Statement

http://imgur.com/UsKsaOn

Homework Equations

Why is the answer in joules multiplied by 2/3 and 1/2? I can follow the rest of the problem.

The Attempt at a Solution

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I get the same solution as my teacher if I use the constants 2/3 and 1/2. I just don't understand where they are coming from.

 

[Divya Shyam Singh]

I would want to correct the information you have given by looking at the solved equation.
The ratio of distribution of engergy for particle 1 and particle 2 is 2:1
So the energy of particle 1 will be: 2/(2+1)= 2/3 times the total energy
Energy of particle 2: 1/(1+2)= 1/3 times the total energy.
This "total energy" is basically the change in the electric potential as the particles move from being 0.1m apart to 0.2m apart.

 

[PurelyPhysical]

I would want to correct the information you have given by looking at the solved equation.
The ratio of distribution of engergy for particle 1 and particle 2 is 2:1
So the energy of particle 1 will be: 2/(2+1)= 2/3 times the total energy
Energy of particle 2: 1/(1+2)= 1/3 times the total energy.
This "total energy" is basically the change in the electric potential as the particles move from being 0.1m apart to 0.2m apart.

Thank you very much! This clears it up for me. It never occurred to me to look at ratios that way. Is there a name for what this particular concept is in math?

 

[Divya Shyam Singh]

Ummm...not really. It is generally covered in the topic of ratios and proportions.
I will explain a bit more here for you to have a better understanding:
Suppose a line segment of length A units is to be divided into a ratio of x:y
Then the length of each unit can be calculated as:
Ax/(x+y)
Similarly, the length of the other part will be:
Ay/(x+y)

To check we can easily see that the sum of each of the part of the line segment should be equal to the total length of the line segment.
so
Ax/(x+y) + Ay(x+y)= A

Hope it helped!

 

[collinsmark]

Thank you very much! This clears it up for me. It never occurred to me to look at ratios that way. Is there a name for what this particular concept is in math?

The energy ratios can be calculated by invoking conservation of momentum.

The total momentum of the system starts out as 0 and ends as 0.

$m_1 \vec {v_1} + m_2 \vec {v_2} = 0.$

Placing the particles on a line and noting that they move in opposite directions, we can get rid of the vector notation.

$m_1 v_1 - m_2 v_2 = 0.$ [Edit: I'm just using the magnitudes of the velocities in this equation. If you'd rather allow negative velocities to indicate direction, then use $m_1 v_1 + m_2 v_2 = 0.$ That's arguably the better approach anyway.]

Substituting $m_2 = 2m_1$ you can calculate a simple relationship between $v_1$ and $v_2$ (you can do that for yourself ).

Note that the energy for each particle is

$E_1 = \frac{1}{2}m_1 v_1^2,$

$E_2 = \frac{1}{2}m_2 v_2^2,$

And the total energy is,

$E_T = E_1 + E_2,$

if you make the appropriate substitutions, you'll find the energy ratios are 2/3 and 1/3 of the total.

 

[PurelyPhysical]

Ummm...not really. It is generally covered in the topic of ratios and proportions.
I will explain a bit more here for you to have a better understanding:
Suppose a line segment of length A units is to be divided into a ratio of x:y
Then the length of each unit can be calculated as:
Ax/(x+y)
Similarly, the length of the other part will be:
Ay/(x+y)

To check we can easily see that the sum of each of the part of the line segment should be equal to the total length of the line segment.
so
Ax/(x+y) + Ay(x+y)= A

Hope it helped!

This does help, thank you very much :D