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.<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

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

2/What was the speed of a 1.8g  bullet if the block's mass is 1.0kg  and if the spring, with k  = 21N/m , was compressed by 17cm  ?

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

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Explanation / Answer

For a system of bullet and block, momentum conservation is applied as no external force is acting on the system,

mv = (M + m) x v' ----------------------------------(a)

For a system of bullet-block and spring, energy conservation is applied,

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

v'^2 = 21 x (0.17)^2 / (1 + 0.0018)

v' = 1.33 m/s

putting the value of v' in equation (a)

Velocity of bullet, v = (1 + 0.0018) x 1.33 / (0.0018)

= 740.21 m/s

Initial Energy of bullet, KE1 = (1/2) m v^2

= (1/2) x 0.0018 x 740.21^(2)

= 493.11 J

Final Enegy of bullet, KE2 = (1/2) m v'^2

= (1/2) x 0.0018 x 1.33^(2)

= 15.9 x 10^(-4)

So, percentage loss in KE of bullet = ([493.11 - 15.9x10^(-4)] / 493.11) X 100

= 99.99 %.

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