Practice Problems

1. A 0.5 kg ball moving at 8 m/s collides head-on elastically with another 0.5 kg ball at rest. Find the velocities of both balls after the collision. Show Solution
   In equal-mass elastic collisions, they exchange velocities. First ball: \( 0 \, \text{m/s} \), Second ball: \( 8 \, \text{m/s} \).
2. A 2.0 kg object moving east at 4 m/s collides inelastically with a 3.0 kg object at rest. Find their common velocity after collision. Show Solution
   \( v = \dfrac{(2)(4)}{2+3} = \dfrac{8}{5} = 1.6 \, \text{m/s east} \).
3. A 1.2 kg cart moving at 2.5 m/s collides elastically with a 2.4 kg cart moving in the opposite direction at 1.5 m/s. Find their velocities after collision. Show Solution
   Using momentum and energy conservation: \( v_1 \approx -2.17 \, \text{m/s} \), \( v_2 \approx 1.53 \, \text{m/s} \).
4. A 0.25 kg ball moving north at 10 m/s explodes into two fragments of 0.15 kg and 0.10 kg. The 0.15 kg piece moves east at 12 m/s. Find the velocity of the other piece. Show Solution
   Total initial momentum = \( (0.25)(10) = 2.5 \, \text{kg·m/s north} \). After explosion: East momentum = \( 0.15(12) = 1.8 \). Let \( v_x, v_y \) be components of smaller mass velocity. By momentum conservation: \( 0.1 v_x = -1.8 \) → \( v_x = -18 \). \( 0.1 v_y = 2.5 \) → \( v_y = 25 \). Magnitude = \( \sqrt{(-18)^2 + 25^2} \approx 30.8 \, \text{m/s} \). Direction: west of north.
5. A 70 kg athlete runs east at 6 m/s and collides inelastically with a 30 kg child at rest. Find their final velocity. Show Solution
   \( v = \dfrac{70(6)}{70+30} = \dfrac{420}{100} = 4.2 \, \text{m/s east} \).