Momentum & Collisions Cheat Sheet

A printable reference covering momentum, impulse, conservation of momentum, elastic collisions, inelastic collisions, and recoil for grades 10-11.

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Momentum and collisions connect force, time, mass, and velocity in moving objects. This cheat sheet helps students quickly identify which quantities are conserved and which formulas apply before, during, and after a collision. It is useful for solving cart, puck, ball, explosion, and recoil problems in one dimension and simple two-dimensional cases. The main idea is that momentum $\vec{p} = m\vec{v}$ is a vector, so direction matters in every setup. The most important rule is conservation of momentum, written as $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$ when the net external impulse is zero. Impulse changes momentum according to $\vec{J} = \Delta \vec{p} = \vec{F}_{\text{avg}}\Delta t$ . Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve momentum but not kinetic energy. Careful sign choices, clear before-and-after diagrams, and consistent units make most collision problems much easier.

Key Facts

Linear momentum is $\vec{p} = m\vec{v}$ , where $\vec{p}$ is measured in $\text{kg}\cdot\text{m/s}$ .
Impulse is the change in momentum, so $\vec{J} = \Delta \vec{p} = m\vec{v}_{f} - m\vec{v}_{i}$ .
For a constant or average force, impulse is $\vec{J} = \vec{F}_{\text{avg}}\Delta t$ .
If net external impulse is zero, total momentum is conserved: $\sum \vec{p}_{i} = \sum \vec{p}_{f}$ .
Kinetic energy is $K = \frac{1}{2}mv^2$ , and total kinetic energy is conserved only in an elastic collision.
For a perfectly inelastic collision where two objects stick together, $v_{f} = \frac{m_{1}v_{1i} + m_{2}v_{2i}}{m_{1} + m_{2}}$ .
The coefficient of restitution in one dimension is $e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}$ , with $e = 1$ for a perfectly elastic collision.
The center-of-mass velocity of a two-object system is $v_{\text{cm}} = \frac{m_{1}v_{1} + m_{2}v_{2}}{m_{1} + m_{2}}$ .

Vocabulary

Momentum: Momentum is a vector quantity equal to mass times velocity, written as $\vec{p} = m\vec{v}$ .
Impulse: Impulse is the product of average force and time interval, and it equals the change in momentum: $\vec{J} = \vec{F}_{\text{avg}}\Delta t = \Delta \vec{p}$ .
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 in which objects stick together and move with one shared final velocity.
System: A system is the set of objects being analyzed, and momentum is conserved for the system when the net external impulse is zero.

Common Mistakes to Avoid

Ignoring direction, because momentum and impulse are vectors. Choose a positive direction and give velocities signs such as $v = +3.0\,\text{m/s}$ or $v = -3.0\,\text{m/s}$ .
Assuming kinetic energy is always conserved, because only elastic collisions conserve $K = \frac{1}{2}mv^2$ . In inelastic collisions, momentum is conserved but some kinetic energy becomes sound, heat, or deformation.
Using mass in grams instead of kilograms, because SI momentum units require $m$ in $\text{kg}$ and $v$ in $\text{m/s}$ . Convert $500\,\text{g}$ to $0.500\,\text{kg}$ before calculating.
Mixing initial and final velocities in the same side of the equation, because conservation compares before and after states. Set up $m_{1}v_{1i} + m_{2}v_{2i} = m_{1}v_{1f} + m_{2}v_{2f}$ clearly.
Forgetting external forces, because momentum is conserved only when net external impulse is zero or negligible. Friction, gravity along a ramp, or a push from outside the system can change total momentum.

Practice Questions

1 A $0.50\,\text{kg}$ cart moving at $4.0\,\text{m/s}$ collides and sticks to a $1.50\,\text{kg}$ cart at rest. Find their shared final velocity.
2 A $0.20\,\text{kg}$ ball changes velocity from $+15\,\text{m/s}$ to $-10\,\text{m/s}$ after hitting a wall. Find the impulse on the ball.
3 A $70\,\text{kg}$ skater at rest throws a $5.0\,\text{kg}$ object forward at $8.0\,\text{m/s}$ . Assuming no friction, find the skater’s recoil velocity.
4 Two identical carts collide on a nearly frictionless track. One collision makes them bounce apart, and another makes them stick together. Explain which collision loses more kinetic energy and why momentum can still be conserved.