Elastic and inelastic collisions are used to predict how objects move before and after they interact. This cheat sheet helps students organize collision problems by identifying the system, choosing a direction, and applying conservation laws correctly. Worked example patterns are useful because most collision questions follow a small number of reliable steps.
Students need these methods for carts, balls, vehicles, explosions, and lab data analysis.
The most important idea is conservation of momentum, written as , when the net external impulse is negligible. Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve momentum but not kinetic energy. In a perfectly inelastic collision, the objects stick together and share one final velocity.
Always keep signs for direction, because velocity and momentum are vector quantities.
Key Facts
- Momentum is calculated with , where is momentum, is mass, and is velocity.
- For an isolated collision system, total momentum is conserved: .
- Kinetic energy is calculated with , and it is always zero or positive.
- In an elastic collision, both and are true.
- In a perfectly inelastic collision, the objects stick together, so .
- The final velocity for a perfectly inelastic collision is .
- Kinetic energy lost in a collision can be found with , and an inelastic collision has .
- In one-dimensional elastic collisions, the relative speed relationship is .
Vocabulary
- Momentum
- Momentum is the quantity of motion of an object, calculated by .
- Impulse
- Impulse is the change in momentum caused by a force acting over time, written as .
- Elastic collision
- An elastic collision is a collision in which both total momentum and total kinetic energy are conserved.
- Inelastic collision
- An inelastic collision is a collision in which total momentum is conserved but total kinetic energy decreases.
- Perfectly inelastic collision
- A perfectly inelastic collision is a collision where objects stick together and move with the same final velocity.
- Isolated system
- An isolated system is a group of objects with no significant net external impulse during the collision.
Common Mistakes to Avoid
- Ignoring direction signs is wrong because velocity and momentum are vectors. Choose one positive direction and use negative velocities for motion in the opposite direction.
- Using is wrong because kinetic energy includes the factor . The correct formula is .
- Assuming kinetic energy is conserved in every collision is wrong because only elastic collisions conserve kinetic energy. In inelastic collisions, some kinetic energy changes into heat, sound, deformation, or internal energy.
- Forgetting that stuck objects share one final velocity is wrong in a perfectly inelastic collision. Use instead of giving each object a separate final velocity.
- Dropping units during calculations is risky because momentum and kinetic energy use different units. Momentum is measured in , while kinetic energy is measured in .
Practice Questions
- 1 A cart moving at collides and sticks to a cart at rest. Find their shared final velocity .
- 2 A ball moving at hits a wall and rebounds at in the opposite direction. Find the ball's change in momentum .
- 3 Two carts collide elastically in one dimension. Cart has and , while cart has and . Predict and .
- 4 A collision conserves momentum but the total kinetic energy after the collision is smaller than before. Explain whether the collision is elastic, inelastic, or impossible, and justify your answer.
Understanding Elastic and Inelastic Collisions Worked Examples
During the brief contact time, each object pushes on the other with forces that are equal in size and opposite in direction. These forces can be very large, but they are internal forces if both objects are included in the system. Their momentum changes cancel when added together.
This is why choosing the system matters so much. If a cart hits a wall, the wall is outside a system containing only the cart.
The wall gives the cart an external impulse, so the cart's momentum alone does not stay constant. If the system includes the cart, wall, and Earth, the accounting becomes more complete, though it is often less practical for a school calculation.
Kinetic energy behaves differently because it can change form during impact. A squashy ball compresses, a bumper bends, surfaces heat up, and sound is produced. Objects may even gain rotational motion that is not included in a simple one-dimensional model.
None of this means energy disappears. It means that energy has moved out of the kinetic energy of straight-line motion. An elastic collision is therefore an ideal case in which the objects return energy from deformation without a net loss of translational kinetic energy.
Hard steel balls and air-track gliders can come close, but real measurements usually show some loss. A collision does not need to involve sticking to be inelastic. Two objects can bounce apart while still losing kinetic energy.
A reliable worked example starts with a direction choice, usually right as positive. Make a small before-and-after table for mass and signed velocity. A leftward velocity must be negative even when the object is moving fast.
Then describe what physically happens. If the objects lock together, they must have one shared final velocity. If they separate, each can have its own final velocity.
Count the unknown values before selecting equations. Momentum gives one relationship.
A truly elastic one-dimensional problem needs a second relationship, often the kinetic energy condition or the relative-speed rule. This step prevents a common mistake where students try to find two unknown final velocities using only one equation.
The answer should be checked against the story of the collision. A light cart striking a much heavier stationary cart often rebounds or slows sharply, while the heavy cart changes speed only a little. If two equal-mass carts collide elastically and one begins at rest, their speeds commonly exchange.
These patterns are useful checks, not replacements for calculation. In lab work, use measured masses and velocities with units, then compare total momentum before and after.
Small differences are expected because of track friction, sensor timing limits, uneven motion, and forces from the surroundings. A result with a large mismatch often points to a missing negative sign, mixed units, or a system that was not isolated well enough.