π₯Inelastic Collisions
When Energy is Not Conserved
An inelastic collision is one where kinetic energy is NOT conserved β some KE is converted to heat, sound, or deformation. Momentum is still conserved!
Perfectly inelastic collision: the objects stick together after the collision (maximum KE loss):
mβvβα΅’ + mβvβα΅’ = (mβ + mβ)v_f
Example: A car rear-ending another and they stick together, clay balls colliding, a bullet embedding in a block.
KE lost = KE_before β KE_after
The KE loss measures how inelastic the collision was.
Think About It
In a perfectly inelastic collision, is momentum conserved? Is kinetic energy conserved? Can we ever have a collision where kinetic energy is gained?
βοΈ Worked Example: Ballistic Pendulum
Problem: A 0.01 kg bullet moving at 400 m/s embeds in a 2 kg block at rest. Find the final velocity of the block+bullet system.
π Key Equations
Inelastic collisions
m_1v_1i + m_2v_2i = (m_1+m_2)v_f (perfectly inelastic)Ξ KE = KE_f - KE_i < 0 (energy is lost)β οΈ Common Mistakes
Misconception: Thinking momentum is NOT conserved in inelastic collisions because energy is lost.
β Correct thinking: Momentum IS conserved in all collisions (elastic and inelastic), as long as there's no net external force. Only kinetic energy is not conserved in inelastic collisions.
Why: Newton's Third Law ensures internal forces cancel. The lost KE goes to heat, sound, and deformation β not to violating momentum conservation.
Misconception: Using the perfectly inelastic formula (one final velocity) for collisions where the objects don't stick together.
β Correct thinking: The formula mβvβi + mβvβi = (mβ+mβ)v_f only applies when the objects stick together (perfectly inelastic). If they separate, use the general momentum equation with two final velocities.
Why: "Perfectly inelastic" specifically means the objects merge into one β maximum possible KE loss while still conserving momentum.
Misconception: Assuming the KE loss equals the energy converted to "force" or "damage."
β Correct thinking: The KE loss (ΞKE = KE_f β KE_i) is the energy converted to internal energy: heat, sound, deformation of materials.
Why: Energy is conserved overall; it just transforms from mechanical KE into non-mechanical forms during the collision.
π Practice Problems
Try these problems. Check your answer when ready.
A 2 kg cart at 6 m/s collides and sticks to a stationary 4 kg cart. What is the final velocity of the combined system?
A 60 kg football player running at 8 m/s tackles a stationary 90 kg player. They move together after the tackle. Find the final speed.
A 5 kg block at rest is hit by a 0.1 kg ball moving at 50 m/s. The ball embeds in the block. Find (a) final velocity, (b) KE before and after, (c) KE lost.
Two lumps of clay collide head-on. Lump A: 3 kg at +4 m/s. Lump B: 2 kg at β5 m/s. They stick together. Find the final velocity and state which direction.
A 1200 kg car moving at 25 m/s rear-ends a stationary 800 kg car. After the collision, the 1200 kg car is moving at 10 m/s. Find the velocity of the 800 kg car and determine how much KE was lost.
Ξ KE = KE_f - KE_iIn a ballistic pendulum experiment, a 10 g bullet embeds in a 990 g block, which swings up to a height of 0.2 m. Find the bullet's initial speed. (g = 10 m/sΒ²)
v_bullet = \frac(m_bullet + m_block)m_bulletβ(2gh)Finished reading through this lesson?