You are at the controls of a particle accelerator, sending a beam of 4.90 ? 10 7

Question

You are at the controls of a particle accelerator, sending a beam of 4.90 ? 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 4.50 ? 107 m/s. Assume that the initial speed of the target nucleus is negligible and the collision is elastic.

(a) Find the mass of one nucleus of the unknown element. Express you answer in terms of the proton mass m.
? m

(b) What is the speed of the unknown nucleus immediately after such a collision?
m/s

Explanation / Answer

From conservation of momentum we have 1.67x10^-27*4.90x10^7 = -1.67x10^-27*4.5x10^7 + M*V

So M*V = 1.67x10^-27*(4.9x10^7 + 4.50x10^7) = 1.5698x10^-19kg-m/s

From conservation of K we have 1/2*m*v1^2 =1/2*m*v2^2 + 1/2*M*V^2

So 1/2*1.67x10^-27*(4.9x10^7)^2 - 1/2*1.67x10^-27*(4.50x10^7)^2 = 1/2*M*V^2

so 1/2*M*V^2 = 3.1396x10^-13

But M*V = 1.5698x10^-19

so 1/2*1.5698x10^-19*V =3.1396x10^-13

V=4.00x10^6m/s

Therefore M = 1.5698x10^-19/4.00x10^6 = 3.92x10^-26kg = 3.92x10^-26/1.67x10^-27kg/proton

= 23.5protons

and again V = 4.0x10^6m/s

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