Question

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass mmm is fired into a block of mass MMM. 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.

What was the speed of a 1.8 gg bullet if the block's mass is 1.2 kg and if the spring, with k = 37 N/mN/m , was compressed by 14 cm ?

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

Answer 1

Mass of the bullet, m = 0.0018 kg

Mass of the block. M = 1.2 kg

Spring constant, k = 37 N/m

Compression of the spring, x = 0.14 m

Work done to compress the spring, W = ½ k x² = ½ x 37 x 0.14 x 0.14= 0.3626 J

Energy of the bullet block system, E = ½ (m+M) V²

Work done to compress the spring is provided by the energy of bullet block system.

½ (m+M) V² = 0.3626

½ x (0.0018 + 1.2 ) V² = 0.3626

The common velocity of bullet block, V = 0.777 m/s

Using law of conservation,

mv = (m+M) V

initial speed of the bullet, v = (m+M) V/m = ( 0.0018 + 1.2 ) x 0.777/0.0018 = 518.7 m/s

speed of bullet, v = 518 .7 m/s

energy of bullet before hitting the block, K1 = 1/2mv² = ½ x 0.0018 x 518.7² = 1076.2 J

energy of bullet after hitting the block, K2 = 1/2mV² = ½ x 0.0018 x 0.777² = 0.00054 J

percent loss of energy of the bullet =(K2 – K1) x100/K1= (0.00054 – 1076.2) x 100/1076= - 610 %