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4 Momentum and Collisions

Learn how momentum and impulse describe motion changes, when momentum is conserved, and how conservation laws distinguish elastic and inelastic collisions.

Momentum and

An object's depends on both its mass and its velocity:

p⃗=mv⃗.\vec p = m\vec v.

Because momentum is a vector, its direction matches the direction of the velocity. In one dimension, choose a positive direction and represent motion in the opposite direction with a negative velocity.

A net force acting over a time interval changes momentum. The resulting is

J⃗=∫titfF⃗net dt=Δp⃗=p⃗f−p⃗i.\vec J = \int_{t_i}^{t_f} \vec F_{\text{net}}\,dt = \Delta\vec p = \vec p_f-\vec p_i.

For a constant or average force, J⃗=F⃗avgΔt\vec J=\vec F_{\text{avg}}\Delta t. For the same change in momentum, increasing the time taken to slow an object reduces the average force. Airbags and padded surfaces help protect people by increasing the time over which they come to a stop.

Example: A ball with mass 0.15 kg0.15\,\text{kg} moves at +20 m/s+20\,\text{m/s} and rebounds at −15 m/s-15\,\text{m/s}. Its momentum change is

Δp=m(vf−vi)=0.15(−15−20)=−5.25 kg⋅m/s.\Delta p = m(v_f-v_i)=0.15(-15-20)=-5.25\,\text{kg·m/s}.

If contact lasts 0.010 s0.010\,\text{s}, the average net force is

Favg=ΔpΔt=−525 N.F_{\text{avg}}=\frac{\Delta p}{\Delta t}=-525\,\text{N}.

The negative sign means the force points opposite to the chosen positive direction.

Takeaway: links a force acting over time to a change in momentum.

When Momentum Is Conserved

Momentum is conserved when a system has no net external , or when external is negligible over the interval considered. Internal forces between objects occur in equal-and-opposite pairs, so they do not change the total momentum of the system.

The can be written as

p⃗initial=p⃗final.\vec p_{\text{initial}}=\vec p_{\text{final}}.

For two objects moving along one line, this becomes

m1v1i+m2v2i=m1v1f+m2v2f.m_1v_{1i}+m_2v_{2i}=m_1v_{1f}+m_2v_{2f}.

Choose a positive direction before substituting velocities, and keep the signs consistent. A collision can involve large forces between the objects while still conserving momentum: those forces are internal if both objects are included in the system.

Example: A 2.0 kg2.0\,\text{kg} cart moving at +3.0 m/s+3.0\,\text{m/s} collides with a 1.0 kg1.0\,\text{kg} cart moving at −1.0 m/s-1.0\,\text{m/s}. If they stick together, gives their shared final velocity:

vf=(2.0)(3.0)+(1.0)(−1.0)2.0+1.0=1.67 m/s.v_f=\frac{(2.0)(3.0)+(1.0)(-1.0)}{2.0+1.0}=1.67\,\text{m/s}.

The positive result shows that the joined carts move in the first cart's original direction.

Takeaway: Check whether external is negligible, define the system, and use signed velocities when applying momentum conservation.

Collision Types and Kinetic Energy

Collisions are classified by what happens to the system's kinetic energy, given by

K=12mv2.K=\frac12mv^2.

In an , both total momentum and total kinetic energy are conserved. In an , momentum is conserved but some kinetic energy is transformed into other forms, such as sound, thermal energy, or deformation. Kinetic energy is transformed, not destroyed.

A occurs when the objects stick together after impact. Their final velocities are equal, and for given initial conditions this collision has the greatest possible loss of kinetic energy.

For a one-dimensional , use both conservation equations:

m1v1i+m2v2i=m1v1f+m2v2f,m_1v_{1i}+m_2v_{2i}=m_1v_{1f}+m_2v_{2f},
12m1v1i2+12m2v2i2=12m1v1f2+12m2v2f2.\frac12m_1v_{1i}^2+\frac12m_2v_{2i}^2=\frac12m_1v_{1f}^2+\frac12m_2v_{2f}^2.

As a useful special case, in a head-on between equal masses, if one object is initially at rest, the objects exchange velocities.

Takeaway: Momentum is conserved in both elastic and inelastic collisions in an isolated system; kinetic energy is conserved only in elastic collisions.

A Reliable Problem-Solving Method

Use this sequence to solve collision and problems:

  1. Define the system and decide whether external during the interaction can be neglected.

  2. Choose coordinate directions and assign signs to all velocities.

  3. Apply momentum conservation in each relevant component.

  4. For an , also apply kinetic energy conservation. For a , set the final velocities equal.

  5. Check units, signs, and whether the result fits the physical situation.

Putting the ideas together: Momentum conservation determines how total motion is shared before and after a collision. The collision type tells you whether kinetic energy supplies an additional equation. is useful when the question instead concerns the force and duration of an interaction that changes momentum.