Elastic Collision problem with atoms.

[hellomister]

Homework Statement

A neutron in a nuclear reactor makes an elastic head-on collision with the nucleus of a carbon atom initially at rest.

(a) What fraction of the neutron's kinetic energy is transferred to the carbon nucleus? (The mass of the carbon nucleus is about 12 times the mass of the neutron.)

Homework Equations

Conservation of momentum: M1V1i+M2V2i=M1V1f+M2V2F
Conservation of Kinetic Energy: 1/2M1V1i^2+1/2M2V2i^2=1/2M1V1F^2+1/2M2V2F^2

The Attempt at a Solution

Since V2i=0
M1V1i=M1V1F+M2V2F

and

1/2M1V1i^2=1/2M1V1F^2+1/2M2V2F^2 <--- 3 unknowns V1i,V1F,V2F so i need 3 equations to solve for them.

(1)(V1i)=(1)(V1F)+(12)(V2F)
V2F=V1i-V1F/12

umm so i was wondering if I am on the right track to solving this problem?

 

[chrisk]

You are on the right track. Since V2i = 0 and using the conservation of momentum and kinetic energy equations one gets

V1F = ((M1 - M2)/(M1 + M2))V1i

V2F = (2M1/(M1 + M2))V1i

You arrive at these equations by rearranging the momentum and energy equations by grouping the initial and final velocities of the first mass on one side of the equation and doing the same on the other side for the second mass. Then divide the rearrange energy equation by the rearranged momentum equation using the difference of two perfect squares factoring i.e. a² - b² = (a + b)(a - b). This gives

V1i - V2i = V2F - V1F

Using the above equation and the rearranged momentum equation the final velocities can be obtained.

 

[thomate1]

Yes, you are on the right track to solving this problem. You have correctly identified the conservation of momentum and kinetic energy equations as the relevant equations to use in this elastic collision problem. However, you are correct in noting that you need three equations to solve for the three unknowns (V1i, V1F, and V2F). You can obtain a third equation by using the fact that the collision is elastic, which means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. This can be written as:

1/2M1V1i^2 = 1/2M1V1F^2 + 1/2M2V2F^2

Substituting in the expression you found for V2F, you can then solve for V1F and V2F in terms of V1i. This will allow you to determine the fraction of the neutron's kinetic energy that is transferred to the carbon nucleus.