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## Homework Statement

A positively charged alpha particle is fired, in a direct line, at a positively charged gold nucleus:

o--> O

o: alpha, O: golden nucleus.

Because of their mutual repulsion, the alpha particle does not actually hit the nuleus: it comes to rest, for a moment, some distance from the nucleus and then recedes. The alpha particle may be considered to have undergone an elastic "collision" with the golden nucleus.The golden nucleus has a mass of about 50 times the alpha particle. Compared to the gold nucleus, after the collision, the alpha particle has

A. more momentum but less kinetic energy

B. more momentum and more kinetic energy.

C. less momentum and less kinetic energy

D. less momentum but more kinetic energy

E. the same momentum but more kinetic energy.

## Homework Equations

Elastic collision: object 2 initially at rest. vi is the initial velocity of the alpha particle. The post-collision velocities are:

v1= (m1 - m2)vi / (m1+ m2)

v2= (2*m2)vi / (m1+ m2)

## The Attempt at a Solution

This problem can be solved in two ways: either by using logic and doing no calculations or doing the calculations based on the elastic collision formulas. Logically, I know that the initial momentum is to the right, therefore post collision the overall momentum will be to the right. So the gold nucleus will definitely have more momentum. The question now remains about the kinetic energy. Logically, one would think that even though the net momentum is to the right, the velocity of the gold nucleus < velocity of alpha, due to the huge mass difference, therefore the KE of the alpha particle would be greater. However, to back this up, I wanted to prove it mathematically but it doesn't work out that way.

Let the initial velocity of the alpha particle be vi.

Velocity of gold nucleus post-elastic-collision:

v2= (2*m2)vi / (m1+ m2)

v2= (2*50m)vi / (m+ 50m)

v2= (100m) vi/ (51m) = 100v/51

Velocity of alpha nucleus post-elastic-collision:

v1= (m1 - m2)vi / (m1+ m2)

v1= (m - 50m)vi / (m+ 50m)

v1= (-49m)vi / (51m)= -49/51 v

From this I can see that the velocity of the alpha particle is much less than the velocity of the gold nucleus, which having the greater mass as well, will have the greater momentum. How is it that the math doesn't work out but logically answer D makes sense? Thanks in advance!