The ballistic pendulum is a popular textbook example of a collision process where both momentum and total mechanical energy are conserved, but during different time intervals. During the interval in which 3. Ruler the steel ball collides inelastically with the retaining cup, linear momentum is conserved (even though mechanical energy is converted to heat):

1. $m_b v_A = (m_b + m_c ) v_B$

where $m_b=$ mass of ball, $m_c=$ mass of cup, $v_A=$ speed of ball before collision, $v_B =$ speed of ball + cup after collision. During the interval in which the ball + cup swing together upward, and eventually come to rest, mechanical energy is conserved:

2. $\frac{1}{2} (m + m_c )v_{B}^2 = (m_b + m_c )gh$

where $h =$ is the change of height as the ball + cup swing upward. Solving the above two equations gives

3. $v_A = (m_b + m_c ) \sqrt{2gh}$

For this demo, typical values are $m_b =$ 68g, $m_c=$ 294g, and $h =$ 7.6cm, yielding $v_A \approx$ 6.5 m/s. This value can be independently measured using $v_A = \sqrt{g R^2/2H}$ where $R =$ range, and $H =$ launch height when the steel ball is projected without the pendulum to intercept it. Typical values are $R = 3\text{m}$ and $H = 1\text{m}$ yielding $v_A \approx$ 6.6 m/s.