(a) the speed of each proton after the collision in terms of v i (b) the directi

Question

(a) the speed of each proton after the collision in terms of vi

(b) the direction of the velocity vectors after the collision (assume that the initially moving proton scatters toward the positive y direction)

initially moving proton ? vi initially at rest proton ? vi A proton, moving with a velocity of vi , collides elastically with another proton that is initially at rest. Assuming that after the collision the speed of the initially moving proton is 2.70 times the speed of the proton initially at rest, find the following. (a) the speed of each proton after the collision in terms of vi initially moving proton ? vi initially at rest proton ? vi (b) the direction of the velocity vectors after the collision (assume that the initially moving proton scatters toward the positive y direction) initially moving proton ½ relative to the +x direction initially at rest proton ½ relative to the +x direction

Explanation / Answer

a)

If the protons collide elastically, then kinetic energy is conserved. Let the mass of the protons be m and the speed of the target proton after the collision be v. In that case,

KEi = KEf

0.5mvi2= 0.5m(2v)22 + 0.5mv2

vi2 = 5v2

vi=v*sqrt(5)

b)

we also know that momentum must be conserved. Since momentum is a vector, it must be conserved in all directions. Consider, then, the direction perpendicular to the initial direction of travel of the projectile proton.

That must be 0. So whatever momentum the projectile proton has after the collision perpendicular to its initial direction of travel must be canceled by the momentum of the target proton perpendicular to the initial direction of travel after the collision.

We know that the momentum of the projectile proton in that direction is

If Q is the direction of the projectile proton (it will have to be a negative angle, because it will have to go off in the opposite direction), then we know it's momentum is

2mv sin(theta) = mv sin(Q)

The m's and v's cancel, so

2 sin(theta) = sin(Q)

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