Question

A 34-g bullet travelling at 120 m/s embeds itself in a wooden block on a smooth surface. The block then slides toward a spring and collides with it. The block compresses the spring (k = 99 N/m) a maximum of 1.2 cm. Calculate the mass of the block of wood. (6 marks)

Answer 1

1) Find the kinetic energy of the block and bullet from the elastic potential energy:

$$\dfrac 12kx^2=\dfrac 12(M+m)v^2,\\\space\\ kx^2=(M+m)v^2.$$

Here, v is the speed of the block and bullet right after the bullet hit the block. This speed can be found from conservation of momentum:

$$mu+M·0=(M+m)v→v=\dfrac{mu}{M+m}.$$

Substitute:

$$kx^2=(M+m)\bigg(\dfrac{mu}{M+m}\bigg)^2.\\\space\\ kx^2=\dfrac{(mu)^2}{M+m}→M=\dfrac{(mu)^2}{kx^2}{}-m,\\\space\\ M=\dfrac{(0.034·120)^2}{99·0.012^2}-0.034=1168\text{ kg}.$$