You have been asked to design a \"ballistic spring system\" to measure the speed

Question

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A spring whose spring constant is k is suspended from the ceiling. A block of mass M hangs from the spring. A bullet of mass m is fired vertically upward into the bottom of the block. The spring's maximum compression d is measured.

A) Find an expression for the bullet's speed V. Express your answer in terms of the variables m, M, k, d, and constant g.

B) What was the speed of a 15g bullet if the block's mass is 2.2kg and if the spring, with k = 51 N/m , was compressed by 50cm.

Explanation / Answer

first do conservation of energy

Ei = Ef

(M + m) g d + 1/2 kd^2 = 1/2 ( M + m) v^2

but we from conservation momentum
m V = ( M + m) v
v = m V/(M + m)

so plug into energy

(M + m) g d + 1/2 kd ^2 = 1/2 (M + m) m^2 V^2/(M+m)^2 = 1/2 m^2 v^2/(M+m)

2 (M + m)^2 g d/m^2 + ( M + m) k d^2/m^2 = V^2
V = sqrt(2 (M + m)^2 g d/m^2 + ( M + m) k d^2/m^2)

b)
V = sqrt( 2 (2.2+15.0E-3)*9.81*.5/(15.0E-3)^2 + (2.2+15.0E-3)*51*.5^2/(15.0E-3)^2)= 471.3 m/s

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