When objects crash together or fly apart, momentum is always conserved — but kinetic energy is not. Whether the energy survives splits collisions into two families: elastic and inelastic. Knowing which is which, and applying conservation of momentum along a single line, lets you solve almost any straight-line collision or explosion.
📘 What you need to know
In both collisions and explosions, momentum is always conserved; kinetic energy may not be
An elastic collision conserves kinetic energy; an inelastic one does not
A totally inelastic collision is the extreme case where the objects stick together
An explosion starts from rest and pushes objects apart — kinetic energy is created from stored energy
In 1D, velocities lie along a single line and can be positive or negative
To test if a collision is elastic, compare the total kinetic energy before and after
Collisions and explosions in one dimension
A collision is when two or more moving objects come together and push on each other for a short time. An explosion is the reverse: objects that start at rest are pushed apart. In one dimension everything happens along a single straight line, so each velocity is just a positive or negative number depending on its direction.
In a collision, moving objects come together along a line. In an explosion, a stationary object bursts into pieces that fly apart. Momentum is conserved in both.
Elastic and inelastic collisions
The key distinction is what happens to kinetic energy:
Elastic KE conserved
vs
Inelastic KE not conserved
In an elastic collision, the total kinetic energy after equals the total before — none is lost. Perfectly elastic collisions are an idealisation: they happen between particles like gas molecules, but never quite in everyday life. In an inelastic collision, some kinetic energy is transferred away — to heat, sound, or deforming the objects — so the total kinetic energy drops. Almost every real, large-scale collision is inelastic.
The extreme case is a totally inelastic collision, where the objects stick together and move off as one. This loses the maximum possible kinetic energy while still conserving momentum.
Elastic: the balls bounce apart and keep all their kinetic energy. Totally inelastic: they stick together, losing kinetic energy to heat, sound and deformation. Both conserve momentum.
Here’s the mental model that keeps it straight: momentum is the rule that never breaks — use it to find unknown velocities in any collision or explosion. Kinetic energy is the test: work it out before and after, and if it’s the same, the collision was elastic; if it dropped, inelastic. Never assume KE is conserved — always check it. And in an explosion, KE actually goes up, released from stored chemical or elastic energy.
Testing for elastic or inelastic
To decide whether a collision is elastic, you compare the total kinetic energy before and after, using:
Kinetic energyEk = ½mv2
🛠️ Solving a collision or explosion
Choose a positive direction and note each mass and velocity with its sign.
Apply conservation of momentum: total p before = total p after — solve for the unknown velocity.
If they stick together, use one combined mass afterwards.
To check elastic vs inelastic, compute total Ek = ½mv2 before and after and compare.
WE 1
A 3.0 kg trolley moving at 4.0 m s−1 collides with a stationary 1.0 kg trolley. They stick together. Find their common velocity, and show that the collision is inelastic.
Step 1 — conserve momentum (they stick)(3.0 × 4.0) + 0 = (3.0 + 1.0) × v
12 = 4.0v → v = 3.0 m s⁻¹Step 2 — kinetic energy beforeEk = ½ × 3.0 × 4.0² = 24 JStep 3 — kinetic energy afterEk = ½ × 4.0 × 3.0² = 18 Jv = 3.0 m s⁻¹; KE drops 24 → 18 J = inelastic6 J of kinetic energy is lost to heat, sound and deformation — but momentum stays at 12 kg m s⁻¹.
WE 2
A rifle of mass 4.0 kg fires a bullet of mass 20 g at 400 m s−1. The rifle and bullet are at rest before firing. Calculate the recoil velocity of the rifle.
Step 1 — total momentum before = 0 (at rest)Step 2 — conserve momentum (right = positive)
0 = (0.020 × 400) + (4.0 × v)
0 = 8.0 + 4.0vStep 3 — solve for the rifle’s velocityv = −8.0 ÷ 4.0v = −2.0 m s⁻¹ (recoil, backward)The minus sign shows the rifle kicks back opposite the bullet. The kinetic energy that appears comes from the propellant — an explosion never conserves KE.
💡 Top tips
Momentum is always conserved — use it first, in every collision and explosion.
Never assume kinetic energy is conserved — test it by comparing before and after.
Totally inelastic = stick together — treat them as one combined mass.
Explosions start from zero momentum, so the pieces carry equal and opposite momenta.
Kinetic energy is a scalar — it’s never negative, since v2 is always positive.
Quick recap: in every 1D collision or explosion, momentum is conserved — use it to find unknown velocities. Kinetic energy is conserved only in elastic collisions; inelastic ones lose it (totally inelastic ones stick together and lose the most). Test elasticity by comparing total Ek = ½mv2 before and after. Explosions create KE from stored energy.
⚠ Common mistakes
Assuming kinetic energy is conserved in every collision — only elastic ones conserve it
Forgetting the sign of velocities — recoiling and rebounding objects go negative
Treating stuck-together objects as separate masses after a totally inelastic collision
Giving kinetic energy a negative value — it’s a scalar and always positive
Thinking momentum is lost when KE is lost — momentum is always conserved
So far every collision has been head-on, along one line. But real collisions — snooker balls, particles in a detector — happen in a plane, with objects glancing off at angles. Momentum is still conserved, but now in two directions at once. That’s next: Collisions & Explosions in Two Dimensions.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
