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Momentum & Collisions cheat sheet - grade 10-11

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Physics Grade 10-11

Momentum & Collisions Cheat Sheet

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

TopicsMomentumNewton's LawsForcesMotionEnergyPhysicsStudy Guide
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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 p⃗=mv⃗\vec{p} = m\vec{v} is a vector, so direction matters in every setup.

The most important rule is conservation of momentum, written as ∑p⃗initial=∑p⃗final\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}} when the net external impulse is zero. Impulse changes momentum according to J⃗=Δp⃗=F⃗avgΔt\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 p⃗=mv⃗\vec{p} = m\vec{v}, where p⃗\vec{p} is measured in kg⋅m/s\text{kg}\cdot\text{m/s}.
  • Impulse is the change in momentum, so J⃗=Δp⃗=mv⃗f−mv⃗i\vec{J} = \Delta \vec{p} = m\vec{v}_{f} - m\vec{v}_{i}.
  • For a constant or average force, impulse is J⃗=F⃗avgΔt\vec{J} = \vec{F}_{\text{avg}}\Delta t.
  • If net external impulse is zero, total momentum is conserved: ∑p⃗i=∑p⃗f\sum \vec{p}_{i} = \sum \vec{p}_{f}.
  • Kinetic energy is K=12mv2K = \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, vf=m1v1i+m2v2im1+m2v_{f} = \frac{m_{1}v_{1i} + m_{2}v_{2i}}{m_{1} + m_{2}}.
  • The coefficient of restitution in one dimension is e=v2f−v1fv1i−v2ie = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}, with e=1e = 1 for a perfectly elastic collision.
  • The center-of-mass velocity of a two-object system is vcm=m1v1+m2v2m1+m2v_{\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 p⃗=mv⃗\vec{p} = m\vec{v}.
Impulse
Impulse is the product of average force and time interval, and it equals the change in momentum: J⃗=F⃗avgΔt=Δp⃗\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 m/sv = +3.0\,\text{m/s} or v=−3.0 m/sv = -3.0\,\text{m/s}.
  • Assuming kinetic energy is always conserved, because only elastic collisions conserve K=12mv2K = \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 mm in kg\text{kg} and vv in m/s\text{m/s}. Convert 500 g500\,\text{g} to 0.500 kg0.500\,\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 m1v1i+m2v2i=m1v1f+m2v2fm_{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. 1 A 0.50 kg0.50\,\text{kg} cart moving at 4.0 m/s4.0\,\text{m/s} collides and sticks to a 1.50 kg1.50\,\text{kg} cart at rest. Find their shared final velocity.
  2. 2 A 0.20 kg0.20\,\text{kg} ball changes velocity from +15 m/s+15\,\text{m/s} to −10 m/s-10\,\text{m/s} after hitting a wall. Find the impulse on the ball.
  3. 3 A 70 kg70\,\text{kg} skater at rest throws a 5.0 kg5.0\,\text{kg} object forward at 8.0 m/s8.0\,\text{m/s}. Assuming no friction, find the skater’s recoil velocity.
  4. 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.

Understanding Momentum & Collisions

A collision is easier to understand when you first choose the system. If the system contains both colliding objects, the contact forces between them are internal forces. During the impact, each object pushes on the other with equal force in the opposite direction.

These forces can be very large, but they act for the same short time and produce opposite momentum changes. Their effects cancel when the two objects are treated as one system.

Forces from outside the system, such as friction from a road or a person holding an object, can change the total momentum. In many classroom collisions, the contact time is so short that outside forces have little effect.

Impulse explains why safety equipment increases stopping time. A seat belt, air bag, helmet, padded landing mat, and bent knees do not remove the needed change in momentum. A moving person still must come to rest.

They make that change happen over a longer time. For the same momentum change, a longer collision time means a smaller average force. Crumple zones in cars work in the same way.

They deform and transfer energy into bending and heating materials. This reduces the force on people inside, though it cannot make a severe crash harmless.

Momentum conservation does not mean that every object keeps its own momentum. It means the total stays fixed for a sufficiently isolated system. One object can lose momentum while another gains it.

In recoil, a gun and bullet begin at rest as a system. After firing, the bullet moves forward and the gun moves backward. The lighter bullet usually has a much greater speed because equal amounts of opposite momentum require different speeds for different masses.

Explosions follow the same pattern. Fragments fly in different directions, yet their vector momenta add to the original total. The center of mass keeps moving at a steady velocity when there is no net external force, even while the objects separate, bounce, or spin.

Kinetic energy needs separate attention because it is not a vector and it cannot cancel by direction. In a collision with sticking, bending, sound, heat, and internal vibration, some kinetic energy becomes other forms of energy. Total energy is still conserved, but the energy of motion is lower afterward.

A ball that rebounds has more kinetic energy after impact than a ball that sticks, though both may obey momentum conservation. When solving problems, draw each object before and after the event. Choose one positive direction and keep every velocity signed.

A leftward velocity is negative if rightward is positive. In two dimensions, conserve momentum separately along horizontal and vertical directions. Do not conserve kinetic energy unless the collision is stated to be elastic or the evidence supports it.