Linear Momentum

Momentum is mass times velocity and is a vector. Impulse is the change in momentum produced by a force over time. A system’s momentum is conserved when its net external impulse is negligible over the interval.

Linear momentum is mass times velocity. Its sign or direction matters, so opposing motions can cancel in the total even when both objects move rapidly. An isolated two-object system can have zero total momentum and substantial kinetic energy. Momentum and kinetic energy therefore encode different information.

Impulse is the integral of force over time, represented by signed area under a force–time graph. For a constant or average force, impulse is force times duration. The net external impulse equals the system’s momentum change.

Total system momentum is conserved when the net external impulse is zero or negligible over the interval. Internal force pairs produce equal and opposite impulses that cancel in the total. Choose the system to include both interacting objects when using collision conservation.

External forces need not be literally absent. During a short horizontal collision, friction may produce a negligible impulse compared with the collision forces. State that approximation rather than claiming gravity and normal force disappear. Over a longer interval, previously negligible forces may matter.

An elastic collision conserves total kinetic energy as well as momentum for the appropriately isolated system. An inelastic collision converts some kinetic energy to internal energy. In a perfectly inelastic collision, objects stick together and share a final velocity.

Momentum can remain conserved even when kinetic energy decreases. Applying kinetic-energy conservation to a sticking collision usually gives an incorrect result. Use momentum first, then compare kinetic energies if asked about energy transformed.

An explosion or spring release can increase system kinetic energy while total momentum remains conserved. If the system begins at rest, its final momenta sum to zero, but the fragments’ speeds need not match. The smaller mass has the larger speed magnitude for two fragments with equal and opposite momenta.

The center of mass moves according to the net external force. Internal interactions can greatly change individual motions without changing center-of-mass motion in an isolated system. This is useful when a system separates into pieces.

For a fixed momentum change, increasing stopping time reduces the magnitude of average force. Cushions and airbags can extend the interaction time and distance. This does not guarantee the same peak force for every force-time shape; average and maximum force are distinct.

A bouncing object can have a greater momentum change than one that merely stops because its velocity reverses. Choose signs and compute final minus initial momentum. A force-time graph may contain both positive and negative regions, which contribute signed impulses.

Area under a force–time graph gives impulse. For a triangular pulse, use one-half base times height; for a rectangular pulse, use base times height. The peak force can be much larger than the average force. The same momentum change over a longer collision time requires a smaller average force, which explains many protective designs.

Choose the sign of the force consistently with the velocity direction. A ball rebounding from a wall changes momentum by more than a ball that merely stops from the same initial speed. Calculate final minus initial momentum with signed velocities rather than subtracting speed magnitudes.

During a short collision, external forces may have negligible impulse even if they matter later. Use momentum conservation for the collision interval, then an energy or force model for subsequent motion. A ballistic-pendulum-style situation, for example, can use momentum during sticking and mechanical energy during the later swing, under the appropriate assumptions.

A perfectly inelastic collision means the objects share a final velocity after sticking. Momentum can be conserved while kinetic energy decreases. In an elastic collision, both momentum and kinetic energy are conserved. Do not impose kinetic-energy conservation merely because the track is frictionless: deformation and internal motion during the collision can still convert kinetic energy.

Internal forces can redistribute motion among parts of a system without changing its total momentum. In an explosion of an initially stationary isolated system, fragments move with momenta that sum to zero. Their kinetic energy can increase because stored internal energy is converted to motion.

The center of mass follows the motion determined by net external force. Two interacting carts can accelerate in opposite directions while the center of mass continues with constant velocity. A larger mass need not have larger momentum after an explosion; for two fragments from rest, their momentum magnitudes are equal while their speed magnitudes are inversely related to mass.

Momentum p=mv is a vector. The impulse theorem Δp=∫F_external dt integrates the net force over a time interval. A large force acting briefly can deliver the same impulse as a smaller force acting longer. Force-time graph height indicates force; signed area indicates momentum change.

Average force is impulse divided by contact duration. If a ball reverses, subtract its signed initial momentum from its final momentum; using only the change in speed misses the reversal. The collision can transfer substantial momentum even when kinetic energy is unchanged.

A system’s total momentum is conserved when its external impulse is negligible. The system may experience external forces during the rest of its motion; what matters during a short collision is the impulse compared with the internal exchange. Gravity can often be neglected during a brief horizontal impact, but that is an approximation to justify.

In two dimensions, conserve both momentum components. Kinetic energy is an additional conserved quantity only for an elastic collision. In a completely inelastic collision, the objects stick and share a velocity. The kinetic energy lost becomes internal energy; it is not missing momentum. Work through the collision first, then use energy or force analysis for subsequent motion.

The center-of-mass velocity is total momentum divided by total mass. In an isolated explosion, pieces can fly apart while the center of mass continues with its original velocity. This allows a check on any proposed fragment velocities before detailed calculation.

A force sensor has a finite sampling rate and may miss a narrow peak. Integrating many data points can still estimate impulse, but clipping the peak biases the area. Compare the integrated impulse with measured momentum change, identify sensor zero offsets, and state sign conventions for both measurements. Repeated impacts should use comparable initial conditions.

Momentum is a vector, so signs and components matter. A system’s momentum changes through external impulse. Internal interactions can be very large during a collision but cancel in the total-system momentum balance. Include both colliding bodies if you want their mutual forces to be internal.

Momentum conservation is an approximation when external impulse is negligible over the chosen interval. A short collision may satisfy this even if gravity acts throughout, because the collision impulse is much larger over that short time. Over a long subsequent slide, friction can produce a substantial external impulse. Do not extend collision conservation automatically to the entire later motion.

In two dimensions, conserve components separately under the appropriate assumptions. Adding speed magnitudes as though all momentum were in one direction loses essential vector information. Sketch the directions before writing the equation.

Impulse is area under force versus time and equals momentum change. A triangular pulse has half the area of a rectangle with the same height and duration. Increasing stopping time for the same momentum change lowers the required average force; it does not necessarily lower peak force by exactly the same factor unless pulse shape is comparable.

In an elastic collision, total kinetic energy is conserved as well as momentum. In an inelastic collision, kinetic energy changes into other forms while momentum may still be conserved for an isolated system. Perfectly inelastic objects stick together, creating a shared final velocity. Conservation of momentum alone does not mean their final kinetic energy equals the initial value.

Recoil and explosions can increase kinetic energy by converting stored internal energy. The system can begin with zero momentum and end with objects moving in opposite directions whose vector momenta sum to zero. Equal momentum magnitudes do not imply equal speeds when masses differ.

The center of mass is a mass-weighted position. More massive objects contribute more strongly to its location. For an isolated system, the center of mass moves at constant velocity even if objects interact internally, rotate or separate.

A collision can radically change each object’s motion while leaving center-of-mass motion smooth. This is a useful consistency check on a calculation. If an initially stationary isolated pair sticks and your result gives net momentum without an external impulse, the model or arithmetic is inconsistent.

Separate stages when a collision is followed by motion under gravity or friction. Use momentum across the short impact, then energy or force methods during the later interval. Kinetic energy may be lost at impact even if mechanical energy is approximately conserved during the subsequent rise.

Total momentum is conserved when net external impulse is negligible over the chosen interval. A short collision can make some external impulses small relative to collision forces, but this requires justification. Internal forces exchange momentum between objects while canceling in the system total.

Kinetic energy is conserved in an elastic collision, not in every momentum-conserving collision. In a perfectly inelastic collision, objects stick together; momentum can remain conserved while kinetic energy becomes deformation, sound, or thermal energy. “Lost kinetic energy” does not mean total energy is destroyed.

For explosions or recoil, initially internal stored energy can become kinetic energy while total momentum remains unchanged. A system initially at rest can split into pieces with equal and opposite momenta but unequal speeds when masses differ.

Impulse equals Δp. A rebound can create a larger momentum change than stopping because velocity reverses sign. Write final minus initial momentum explicitly rather than subtracting speed magnitudes. Average force is impulse divided by contact time, and different force histories can have the same impulse.

The center of mass moves according to net external force. Internal rearrangements do not accelerate the system center of mass in the absence of external force. In one dimension, its position is the mass-weighted average, not the simple midpoint unless masses are equal.

An experimental collision analysis should compare total momentum before and after with uncertainty and external interactions considered. A small discrepancy may reflect measurement limits; a systematic discrepancy can suggest an omitted external impulse. Do not infer exact conservation solely from rounded numbers.