You have been asked to design a \"ballistic spring system\" tomeasure the speed

Question

You have been asked to design a "ballistic spring system" tomeasure the speed of bullets. A spring whose spring constant isk is suspended from the ceiling. A block of massM hangs from the spring. A bullet of mass m isfired vertically upward into the bottom of the block. The spring'smaximum compression d is measured.    a.) Find an expression for the bullet'sspeed vb interms of m, M, k, and d.    b.) What was the speed of a 10 g bullet ifthe block's mass is 2.0 kg and if the spring, with          k =50 N/m, was compressed by 45 cm? *Please type out step-by-step solutions and be thorough. Thiswill help me out tremendously in how to set up this problem andother problems in the future in order to solve it in the mostefficient manner. THANK YOU!* You have been asked to design a "ballistic spring system" tomeasure the speed of bullets. A spring whose spring constant isk is suspended from the ceiling. A block of massM hangs from the spring. A bullet of mass m isfired vertically upward into the bottom of the block. The spring'smaximum compression d is measured.    a.) Find an expression for the bullet'sspeed vb interms of m, M, k, and d.    b.) What was the speed of a 10 g bullet ifthe block's mass is 2.0 kg and if the spring, with          k =50 N/m, was compressed by 45 cm? *Please type out step-by-step solutions and be thorough. Thiswill help me out tremendously in how to set up this problem andother problems in the future in order to solve it in the mostefficient manner. THANK YOU!*

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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