Momentum, Impulse, and Collisions
Students use momentum conservation and impulse to analyze collisions, recoil, and the design of safety technologies.
Illustrations are auto-generated and may be placeholders. They can be refreshed to match the narration.
Momentum as a Vector
Momentum describes the motion of an object and is defined by p = mv, where p is momentum, m is mass, and v is velocity. Its SI unit is kilogram-meter per second. Because velocity is a vector, momentum has both magnitude and direction. A sign convention helps represent direction mathematically; for example, motion to the right may be positive and motion to the left negative. Consider a 1,200-kilogram car moving east at 15 meters per second. Its momentum is 18,000 kilogram-meters per second east. A 2,400-kilogram truck moving west at 10 meters per second has momentum of 24,000 kilogram-meters per second west, or negative 24,000 kilogram-meters per second under this sign convention. Comparing velocity alone is insufficient because mass also determines momentum.
Impulse and Force-Time Graphs
Impulse is the change in an object’s momentum: J = Δp = FΔt when the force is constant. More generally, impulse equals the area under a force-versus-time graph. This relationship explains how force and collision time can trade off while producing the same momentum change. Suppose a ball initially at rest receives a constant force of 50 newtons for 0.20 second. The impulse is 10 newton-seconds, so the ball’s momentum changes by 10 kilogram-meters per second in the force’s direction. On a graph, this impulse is the rectangular area with height 50 newtons and width 0.20 second. A smaller force acting for a longer time could provide the same impulse. In real impacts, force often rises and falls, so the area under a curved graph must be estimated or calculated.
Conservation of Momentum
In an isolated system, the vector sum of momentum remains constant because external impulse is zero or negligible. Internal forces may transfer momentum between objects, but they occur in equal and opposite pairs and do not change the system’s total momentum. For two objects, the relationship is m1v1i + m2v2i = m1v1f + m2v2f. For example, a stationary 60-kilogram skater throws a 3-kilogram backpack to the right at 8 meters per second. The backpack gains 24 kilogram-meters per second of momentum. To keep total momentum at zero, the skater gains 24 kilogram-meters per second to the left, giving the skater a velocity of negative 0.40 meter per second. This recoil example supports momentum conservation only when the skater-backpack system experiences negligible external horizontal force.
Elastic and Inelastic Collisions
Total momentum is conserved in both elastic and inelastic collisions when external impulse is negligible, but kinetic energy behaves differently. In an elastic collision, total kinetic energy is also conserved. In an inelastic collision, some kinetic energy becomes thermal energy, sound, or deformation. If objects stick together, the collision is perfectly inelastic. For example, a 1.0-kilogram cart moving right at 4.0 meters per second strikes a stationary 1.0-kilogram cart and sticks to it. Initial momentum is 4.0 kilogram-meters per second. Using 4.0 = (2.0)vf gives a shared final velocity of 2.0 meters per second to the right. Initial kinetic energy is 8.0 joules, while final kinetic energy is 4.0 joules. Momentum is conserved, but 4.0 joules is transformed into other forms of energy.
Engineering Safer Collisions
Safety technologies reduce injury by increasing collision time, spreading force, or absorbing kinetic energy. For the same change in momentum, Favg = Δp/Δt shows that a longer stopping time produces a smaller average force. Seat belts stretch slightly, airbags compress, helmets deform, and vehicle crumple zones collapse in controlled ways. For example, stopping a passenger with a momentum change of 600 kilogram-meters per second in 0.030 second would require an average force of 20,000 newtons. Extending the stopping time to 0.15 second reduces the average force to 4,000 newtons. Engineers evaluate test data, material limits, cost, and human biology when proposing safety standards. Students and citizens should also evaluate whether laws, enforcement systems, and access to approved safety equipment use reliable evidence and protect different communities fairly.