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 bullet of mass m is fired into a block of mass M. The block, with the embedded bullet, then slides across a frictionless table and collides with a horizontal spring whose spring constant is k. The opposite end of the spring is anchored to a wall. The spring's maximum compression d is measured. Find an expression for the bullet's speed vB in terms of m, M, k, and d. What was the speed of a 5.6 g bullet if the block's mass is 2.2 kg and if the spring, with k=53N/m, was compressed by 11cm? What percentage of the bullet's energy is "lost"? You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass m is fired into a block of mass M. The block, with the embedded bullet, then slides across a frictionless table and collides with a horizontal spring whose spring constant is k. The opposite end of the spring is anchored to a wall. The spring's maximum compression d is measured. Find an expression for the bullet's speed vB in terms of m, M, k, and d. What was the speed of a 5.6 g bullet if the block's mass is 2.2 kg and if the spring, with k=53N/m, was compressed by 11cm? What percentage of the bullet's energy is "lost"? Find an expression for the bullet's speed vB in terms of m, M, k, and d. What was the speed of a 5.6 g bullet if the block's mass is 2.2 kg and if the spring, with k=53N/m, was compressed by 11cm? What percentage of the bullet's energy is "lost"?

Explanation / Answer

let v be the initial bullet speed before impact with the block

conservation of momentum:

mv = (M+m)v ----------------------> (i)


Conservation of energy: (k.e of the embedded block is stored as p.e in the spring)

½(M+m)v² = ½kd²

from (i) => v = mv / (M+m)

=> ½m²v²/(M+m) = ½kd²

=> v = d[k(M+m)]/m

plugging in the values

=> v = (0.11)[53(2.2056)]/(0.0056)

v = 212.37 m/s ............Ans.

ei = m1v1^2/2
vf = m1v1/(m1+m2)
ef = (m1+m2)vf^2/2 = (m1+m2)(m1v1/(m1+m2))^2/2 = (m1v1)^2/(2(m1+m2))
ei/ef = m1v1^2/2 * 2(m1+m2) / (m1v1)^2
= m1v1^2 * (m1+m2) / (m1v1)^2
= (m1+m2) / m1
Fraction lost is (ei-ef)/ei = 1-ef/ei = 1-m1/(m1+m2)
Percentage lost is 100*(1-m1/(m1+m2)) = 100*(1-0.0056/2.2056) =99.74%

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