What is the speed of the unknown nucleus immediately after such a collision?Find

Question

What is the speed of the unknown nucleus immediately after such a collision?Find the mass of one nucleus of the unknown element. Express your answer in terms of the proton mass m .

You are at the controls of a particle accelerator, sending a beam of 3.00×107 m/s protons (mass m ) at a gas target of an unknown element. Your detector tells you that some protons bounce straight back after a collision with one of the nuclei of the unknown element. All such protons rebound with a speed of 2.70×107 m/s . Assume that the initial speed of the target nucleus is negligible and the collision is elastic.

Explanation / Answer

Let be v1 the initial speed of the protons, v2 their final speed, V the final velocity of the nuclei, m the mass of the protons, and M the mass of the nuclei.
Momemtum conservation:
m v1 = M V - m v2
Energy conservation:
½ m v1² = ½ m v2² + ½ M V²
With a little algebra, we get
m (v1 + v2) = M V
m (v1² - v2²) = M V²
Dividing m.t.m.
v1 - v2 = V = 3*10^7-2.7*10^7=3*10^6m/s
Solving the first equation for M, and replacing V we obtain:
M = m (v1+v2)/(v1-v2)

M=(m(3*10^7+2.7*10^7))/(3*10^6)= 19*m kg

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