Chapter 8, Problem 65GP

Forensic science. Forensic scientists can measure the muzzle velocity of a gun by firing a bullet horizontally into a large hanging block that absorbs the bullet and swings upward. (See Figure 8.50.) The measured maximum angle of swing can be used to calculate the speed of the bullet. In one such test, a rifle fired a 4.20 g bullet into a 2.50 kg block hanging by a thin wire 75.0 cm long, causing the block to swing upward to a maximum angle of 34.7° from the vertical. What was the original speed of this bullet?

Figure 8.50

Problem 65.

8.65. Set Up: Apply conservation of momentum to the collision and conservation of energy to the motion after the collision. Let +x be to the right.

Solve: Collision: P_i,x = P_f,x gives (4.20 × 10⁻³kg)v = (4.20 × 10⁻³kg + 2.50 kg)v′

Motion after the collision: The kinetic energy of the block immediately after the collision is converted entirely to gravitational potential energy at the maximum angle of swing. The figure below shows that the maximum height h the block swings up is given by h = (0.750 m)(l − cos34.7°) = 0.133 m.

Conservation of energy gives $\frac{1}{2}{m}_{\text{tot}}{v^{\prime }}^2=m_{\text{tot}}gh$ and $v^{\prime }=\sqrt{2gh}=1.61m/s.$

Then the conservation of momentum equation gives

$v=\left(\frac{4.20\times {10}^{-3}\text{kg}+2.50\text{kg}}{4.20\times {10}^{-3}\text{kg}}\right)(1.61\text{m/s})=960\text{m/s}$

Reflect: The original speed of the bullet is nearly three times the speed of sound in air. Kinetic energy is not conserved in the collision. Our analysis assumes that the block moves very little during the collision, while the bullet is coming to rest relative to the block. This is a good approximation since the velocity the block gets in the collision is much less than the initial velocity of the bullet.