All collisions in isolated systems conserve momentum. The question that divides them into types is: what happens to kinetic energy?
When objects stick together after a collision, they share a single final velocity. This is the easiest collision type to solve because momentum conservation gives you one equation with one unknown:
Solving for the final velocity:
Notice that vf is exactly the center-of-mass velocity from Lesson 4.3. This makes sense — after sticking, the system moves as one object at v_cm.
A 0.01 kg bullet moving at 400 m/s embeds itself in a 1.99 kg stationary block on a frictionless surface. Find the final velocity of the bullet-block system.
In a general inelastic collision objects do not stick — they separate after the impact with different velocities. Momentum is still conserved, but KE is not. You have one equation (momentum conservation) and typically one unknown final velocity if the other is given.
The kinetic energy lost goes into internal energy of the system — thermal energy in the deformed materials, sound waves radiating away, and permanent deformation of the objects. This energy leaves the mechanical system but total energy of the universe is still conserved.
Car A (1200 kg) moves right at 15 m/s and rear-ends Car B (1500 kg) moving right at 8 m/s. After the collision Car A moves right at 9 m/s. Find Car B final velocity and the kinetic energy lost.
Elastic collisions conserve both momentum and kinetic energy. This gives you two equations for two unknown final velocities — a solvable system.
Solving this system algebraically (substituting the momentum equation into the energy equation) yields the elastic collision formulas for final velocities:
Set masses, initial velocities, and Object 1 final velocity. The simulator calculates Object 2 final velocity from momentum conservation and classifies the collision from the KE comparison.
Drag v1f toward the elastic value to watch DKE approach zero. When p is conserved AND KE is conserved, v1f has exactly one correct value.
A 3 kg ball moving right at 8 m/s collides elastically with a 1 kg ball at rest. Find the final velocities of both balls.
In any inelastic collision, the kinetic energy lost to internal energy can be calculated directly:
DKE is always negative for inelastic collisions — energy leaves the mechanical system. Zero for elastic — nothing is lost. Never positive in a simple collision — you cannot create kinetic energy from nothing.
A 4 kg object moving at 10 m/s collides with a 6 kg stationary object. Calculate the kinetic energy for both perfectly inelastic and elastic outcomes, and find the energy lost in each case.
Unit 4 complete. Progress Check 4 covers all four lessons — momentum, impulse, conservation, and collisions. Collision FRQs require a written justification of collision type, not just a numerical answer.