A 2 kg ball moving at 4 m/s collides with a stationary 2 kg ball and they stick together. A student applies energy conservation and calculates the final velocity as 4 m/s. What is wrong?
AThe student forgot to include the gravitational potential energy in the calculation
BKinetic energy is not conserved in a perfectly inelastic collision — momentum conservation gives v_f = (2×4 + 2×0)/(2+2) = 2 m/s, and the missing kinetic energy was converted to heat and deformation
CThe calculation is correct — energy conservation always applies to collisions
DThe student should have used the reduced mass, not the total mass
Kinetic energy conservation only holds for elastic collisions. When objects stick together (perfectly inelastic), kinetic energy is not conserved. Applying momentum conservation: total momentum before = 2×4 = 8 kg·m/s; total mass after = 4 kg; so v_f = 8/4 = 2 m/s. Kinetic energy before = ½(2)(4²) = 16 J; kinetic energy after = ½(4)(2²) = 8 J. The 8 J difference was converted to deformation, heat, and sound — it is not lost from the universe, but it left the mechanical energy budget.
Question 2 Multiple Choice
Before a car crash, the total momentum of the two-car system is 8000 kg·m/s and total kinetic energy is 200,000 J. After the crash, momentum is still 8000 kg·m/s but kinetic energy is 150,000 J. A student argues this is impossible because energy must be conserved. Is the student right?
AYes — both momentum and total energy must be conserved, so kinetic energy cannot have decreased
BNo — momentum conservation is satisfied (8000 = 8000), and the 50,000 J reduction in kinetic energy is expected in an inelastic collision; that energy went into deforming metal, generating heat, and producing sound
CYes — momentum is conserved but kinetic energy should also be conserved because no external forces acted
DNo — kinetic energy always increases slightly in real collisions due to the release of chemical energy in the materials
Total energy is always conserved, but kinetic energy is not a conserved quantity in inelastic collisions. The 50,000 J of 'missing' kinetic energy was converted into other forms — deformation energy (crumpled metal), thermal energy (heat generated by friction and deformation), and acoustic energy (the sound of the crash). None of this violates any conservation law. The error is equating 'kinetic energy conservation' with 'energy conservation'; the former is a special condition, the latter is universal.
Question 3 True / False
In any collision between two objects with no net external forces, the total momentum of the system is the same before and after, regardless of whether the collision is elastic or inelastic.
TTrue
FFalse
Answer: True
Momentum conservation follows directly from Newton's third law: the forces the two objects exert on each other are equal and opposite, so the impulses they exchange cancel, leaving total momentum unchanged. This applies to every type of collision — perfectly elastic, partially inelastic, perfectly inelastic, or anything in between — as long as no external forces (like friction with the ground or an applied push) act during the collision. This universality is why momentum conservation is the starting equation for every collision problem.
Question 4 True / False
In a perfectly inelastic collision, most of the kinetic energy before the collision is converted to heat and deformation — so kinetic energy after equals zero.
TTrue
FFalse
Answer: False
If kinetic energy after were zero, the objects would be stationary after the collision — but that would require zero final momentum, violating momentum conservation (unless the initial momentum was also zero). A perfectly inelastic collision is one where the objects stick together and move with a common final velocity v_f = (m₁v₁ + m₂v₂)/(m₁ + m₂). This final velocity is generally nonzero, so the system retains kinetic energy. The perfectly inelastic collision is the *maximum* energy loss case, but the maximum is not 100% unless the initial total momentum happened to be zero.
Question 5 Short Answer
Why can't you solve every collision problem by using conservation of energy, even though energy is always conserved overall? What is the key distinction?
Think about your answer, then reveal below.
Model answer: Total energy is always conserved, but kinetic energy is only conserved in elastic collisions. In inelastic collisions, some kinetic energy is converted to heat, sound, and deformation — forms of energy that leave the mechanical energy budget. When a problem says 'the objects collide' without specifying elastic, you cannot assume kinetic energy is conserved, and using that assumption will give a wrong answer (as in the 4 m/s example). Momentum conservation, by contrast, holds universally because it follows from Newton's third law alone. The key distinction: kinetic energy conservation is a special condition; momentum conservation is a universal consequence of Newton's laws.
This is one of the most important conceptual distinctions in introductory mechanics. Many students try to use both conservation laws in every collision, but the problem is overdetermined for inelastic cases — the two equations together with 'objects stick' give a unique solution only because KE conservation is dropped. The discipline of identifying which conservation laws apply in a given situation is the core skill.