Section 4.4: Elastic Collision

In an elastic collision, both momentum and kinetic energy are conserved. Two objects collide and bounce off each other without any loss of kinetic energy.

\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \] \[ \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \]

Here: \( m_1, m_2 \) are masses, \( v_{1i}, v_{2i} \) initial velocities, \( v_{1f}, v_{2f} \) final velocities.

Example 1

Two carts, 2 kg and 3 kg, move toward each other at 4 m/s and -2 m/s, respectively. Find their velocities after an elastic collision.

Using conservation of momentum and kinetic energy, solving gives:
\( v_{1f} = -1.6 \, \text{m/s}, \quad v_{2f} = 2.4 \, \text{m/s} \)

Example 2

A 1 kg ball moving at 5 m/s collides elastically with a 2 kg ball at rest. Find their final velocities.

After solving:
\( v_{1f} = 1.67 \, \text{m/s}, \quad v_{2f} = 3.33 \, \text{m/s} \)

Practice Problems

A 0.5 kg ball at 6 m/s collides elastically with a 0.5 kg ball at rest. Find final velocities.
Two cars, 1000 kg and 1500 kg, collide elastically head-on. If their velocities before collision are 10 m/s and -5 m/s, find velocities after collision.
A 2 kg puck at 3 m/s collides elastically with a 3 kg puck at 2 m/s. Find final velocities.
A 1.5 kg object moving at 4 m/s hits a 2 kg object at rest. Determine their velocities post-collision.
Two identical spheres move at 2 m/s and -3 m/s. Find velocities after elastic collision.