A 2 kg ball moving at 6 m/s strikes a stationary 2 kg ball in a perfectly inelastic collision. Which principle correctly determines the post-collision velocity?
AKinetic energy conservation: ½(2)(6²) = ½(4)v², so v = 4.24 m/s
BMomentum conservation: (2)(6) = (4)v, so v = 3 m/s
CThe coefficient of restitution alone: e = 0 means v = 0 m/s
DBoth momentum and energy conservation must hold simultaneously: v = 6 m/s
In a perfectly inelastic collision (e = 0), the objects move together after impact. Momentum conservation gives (2)(6) + 0 = (2+2)v, so v = 3 m/s. Option A incorrectly applies kinetic energy conservation — kinetic energy is NOT conserved in inelastic collisions. Option C misapplies e = 0 alone; you still need momentum conservation to find the actual velocity. Option D is wrong because energy conservation cannot be applied to kinetic energy here. The kinetic energy loss is ½(2)(6²) − ½(4)(3²) = 36 − 18 = 18 J, converted to heat, sound, and deformation.
Question 2 Multiple Choice
Two objects collide with coefficient of restitution e = 0.6. The relative velocity of approach was 10 m/s. What is the relative velocity of separation after impact?
A10 m/s — momentum conservation requires equal approach and separation speeds
B6 m/s — the coefficient of restitution gives e × (relative approach) = relative separation
C4 m/s — the kinetic energy lost equals (1−e) of the initial kinetic energy
D0.6 m/s — the restitution coefficient directly gives the post-impact velocity
The coefficient of restitution is defined as e = (relative velocity of separation) / (relative velocity of approach) = (v₂' − v₁') / (v₁ − v₂). With e = 0.6 and approach speed 10 m/s, the relative separation speed is 0.6 × 10 = 6 m/s. This is the second equation needed alongside momentum conservation to solve for both post-impact velocities. Option A would only be true for a perfectly elastic collision (e = 1).
Question 3 True / False
Momentum is conserved in all collisions, regardless of whether the collision is elastic or inelastic.
TTrue
FFalse
Answer: True
Momentum conservation applies to ALL collisions because the internal forces between colliding objects are equal and opposite (Newton's third law), contributing zero net impulse to the system. External forces (gravity, friction) also act during a collision, but the collision duration is so brief that their impulse is negligible. Momentum conservation is exact and universal for collisions. Kinetic energy conservation, by contrast, only holds for perfectly elastic collisions (e = 1) — it is the special case, not the rule.
Question 4 True / False
In a perfectly inelastic collision, most kinetic energy is lost.
TTrue
FFalse
Answer: False
A perfectly inelastic collision (e = 0) means the objects move together after impact — maximum deformation and maximum kinetic energy loss for given initial conditions. However, unless one object was already stationary in the center of mass frame, the combined mass is still moving, and some kinetic energy remains. Total energy is conserved (thermodynamics), but it's redistributed: the kinetic energy that disappears becomes heat, sound, and deformation energy. Only in the special case where both objects have equal mass and one is initially stationary does all kinetic energy convert to other forms.
Question 5 Short Answer
Why is the coefficient of restitution needed in addition to momentum conservation to solve a two-body collision problem?
Think about your answer, then reveal below.
Model answer: A two-body collision has two unknown post-impact velocities. Momentum conservation provides one equation: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'. This alone is underdetermined — infinitely many pairs (v₁', v₂') satisfy it. The coefficient of restitution provides a second independent equation: e = (v₂' − v₁') / (v₁ − v₂), constraining the relative velocity of separation. Together, the two equations uniquely determine both post-impact velocities. The restitution coefficient encodes the material's elastic properties — how 'bouncy' the contact surfaces are — and this physical property is what closes the system.
Without the restitution equation, you know total momentum but cannot determine how it is distributed between the two objects. Any distribution that conserves momentum is mathematically valid; the physics of the materials determines which one actually occurs.