Physics 2.23 - 2.26 - Momentum and collisions

Momentum in collisions

Higher tier only

Momentum is the product of an object's mass and velocity. Because velocity has direction, momentum is also a vector.

Choose a positive direction before calculating. A negative momentum does not mean "less momentum"; it means that the momentum points opposite to the chosen positive direction. A stationary object has zero velocity and therefore zero momentum.

Worked example: signed momentum

A $0.80\,\mathrm{kg}$ trolley moves at $3.5\,\mathrm{m\,s^{-1}}$ to the left. Choose right as positive.

The trolley's momentum has magnitude $2.8\,\mathrm{kg\,m\,s^{-1}}$ and points left. The negative sign records direction.

A collision as a system

For a collision, define the system to include both colliding objects. If the resultant external force on this system is negligible during the short collision, the total momentum of the system is conserved:

During contact, trolley A pushes trolley B and trolley B pushes trolley A with equal forces in opposite directions. The forces act for the same contact time, so the trolleys have equal-magnitude, opposite changes in momentum. One trolley's momentum can change substantially, but the changes add to zero for the two-trolley system.

The third-law forces do not cancel on either trolley because they act on different objects. They explain why the total momentum of the pair remains unchanged. Equal forces also do not require equal changes in velocity: different masses can undergo different accelerations.

Worked example: two trolleys join

Trolley A has mass $0.40\,\mathrm{kg}$ and moves right at $6.0\,\mathrm{m\,s^{-1}}$. It hits stationary trolley B of mass $1.6\,\mathrm{kg}$. They join and move together. Find their shared velocity. Take right as positive and assume the resultant external force on the two-trolley system is negligible during the collision.

Initial momenta:

Total momentum before is $+2.4\,\mathrm{kg\,m\,s^{-1}}$. After the collision, the combined mass is:

Apply conservation of momentum:

The direction is sensible because the system's initial momentum was to the right. The shared speed is below A's initial speed because the same total momentum is now carried by a larger moving mass.

The same momentum accounting can describe trolleys that bounce apart or a moving trolley that catches a stationary one. Momentum conservation concerns the total for the system, not whether the objects join.

Testing the model with trolleys

This is a useful evidence investigation, not a mandatory core practical. Its purpose is to compare total signed momentum immediately before and after a one-dimensional trolley collision.

• Arrangement: use two dynamics trolleys on a level, low-friction track; measure each complete trolley mass with a balance; place a metre scale in the plane of motion; and fix a high-frame-rate camera perpendicular to the track. End stops prevent trolleys leaving the track.
• Variables: the independent variable is trolley A's incoming velocity, varied using at least five marked release positions on a short ramp that ends before the level collision region. The dependent variable is $p_{\text{total, after}}-p_{\text{total, before}}$. Controlled variables are both trolley masses, collision attachments, track angle, release method and camera position.
• Measurements: obtain each signed velocity from displacement divided by time over a short interval immediately before or after contact. Repeat each release setting at least three times. Investigate a trace that is inconsistent with the repeats before deciding whether it is anomalous.
• Processing: calculate $mv$ for each trolley, with one direction positive; add the two values before and after; then compare the difference with the measurement resolution. A graph of total momentum after against total momentum before should be close to a straight $y=x$ relationship if the model is supported.
• Safety: use low speeds, secure the camera and metre scale, fit end stops, and keep hands and faces away from the line of travel.
• Evaluation: track friction or a slight slope creates an external force, so level the track and analyse velocities close to the collision. Parallax and frame timing limit velocity precision, so keep the camera perpendicular, put the scale beside the path and use a high frame rate. Trolley rotation makes a one-dimensional model less valid, so align the trolleys and collision attachments carefully.