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.

1.)Find an expression for the bullet's initial speed vb in terms of m,M ,k , and d.

2.)What was the speed of a 1.5g bullet if the block's mass is 1.2 kg and if the spring, with k = 49 n/m , was compressed by 17 cm ?

3.)What percentage of the bullet's energy is "lost"?

Explanation / Answer

I assume this is about the energy lost in the collision, not whatever happens with the spring (which should conserve energy anyway). 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))

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