4.5 Two-Dimensional Collisions

Solve glancing collisions by conserving momentum independently in perpendicular directions.

Estimated time: 20 minutes

Resolve Momentum into Components

In glancing collisions, bodies leave along different directions, so one equation is not enough. Choose axes, then write separate momentum conservation equations for x and y components. This converts one vector statement into two scalar equations.

$\sum p_{x,\text{before}} = \sum p_{x,\text{after}}, \qquad \sum p_{y,\text{before}} = \sum p_{y,\text{after}}$

Apply trig factors to component velocities after collision.

Workflow for Unknown Speeds and Angles

Start by defining known directions and marking each velocity angle relative to one axis. Build x and y equations symbolically before plugging values. If two unknowns remain, solve simultaneously. If kinetic-energy behavior is requested, compute total kinetic energy before and after only after momentum variables are settled.

A common check is geometric: component signs should agree with the drawn directions. If an answer gives a negative speed, your direction assumption likely needs reversal; keep the magnitude positive and adjust direction wording.

Fast Solve Pattern for Glancing Collisions

After writing component equations, isolate terms of the same unknown velocity and angle. Dividing the y-equation by the x-equation often removes the speed first, giving an angle quickly; then substitute back to get speed. This mirrors the worked-example flow in the chapter source.

$\tan\theta = \frac{p_{y,\text{after (known terms adjusted)}}}{p_{x,\text{after (known terms adjusted)}}}$

A ratio step frequently eliminates one unknown and reduces algebra load.

Test Yourself

In a 2D collision with no external force, which statement is correct?

4.4 Applications: Recoil, Transport, and Safety

Chapter 5: Rigid Body Mechanics Overview