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 mb is fired vertically upward into the bottom of the block and comes to a stop inside the block. The spring's maximum compression d is measured.

(a) Find an expression for the bullet's initial speed Vbi in terms of mb , M , k , and d.

(b) What was the speed of a 10 g bullet if the block's mass is 2.0 kg and if the spring, with k = 50 N/m, was compressed by 45 cm? (bullets generally have speeds greater than 100 m/sec).

Explanation / Answer

a) Use the Conservation of Energy

E1 = E2

K1 + U1 = K2 +U2

Kblock,i + Kbullet,i + Uelas,i =Kblock,f + Kbullet,f +Uelas,f

1/2mVb2 (block isn'tmoving) + 1/2MVb2 +1/2fd2 (uncompressed) =1/2(M+m)Vf2(block + bullet stop moving) +1/2kd2

Vb =?[kd2/M]

b) Vb =?[(50kg*m/s2/m)(0.452m2)/(0.010kg)= 0.318 m/s

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