Momentum in two directions

How do momentum components balance on a tabletop?

A puck moving east hits another puck on a smooth tabletop. Afterward, one moves partly north. If the initial northward momentum was zero and outside horizontal impulses are negligible, the other puck must carry a matching southward momentum component.

A useful starting point: Vector components →

Words and symbols before equations

Component

The signed part of a vector along a chosen axis.

x and y axes

Here +x is east and +y is north, both in the tabletop plane.

Angle θ

Measured counterclockwise from +x for the first outgoing momentum.

Sine/cosine

Ratios giving perpendicular/parallel components: p_y=p sinθ and p_x=p cosθ.

Connect the picture to the physics

Draw the initial and final vectors in the same frame. Momentum is conserved separately along any axis with negligible net external impulse: P_xi=P_xf and P_yi=P_yf. Never add perpendicular magnitudes as if they were along one line.

Suppose initial total momentum is (6,0) kg·m/s and final momentum of A is (2,3). Then B must have (6−2,0−3)=(4,−3) kg·m/s. The y components cancel while the x components add to 6.

For an outgoing vector of magnitude q at angle θ, A contributes (q cosθ,q sinθ). B must make up the remaining components. Increasing θ at fixed q up to 90° reduces A’s eastward component and increases its northward component.

This lesson uses component setup and simple known-component subtraction. Full two-dimensional simultaneous collision solving is outside this AP Physics 1 treatment. Component balance alone also does not establish the kinetic-energy outcome.

A worked example, step by step

An initial total momentum of (8,0) kg·m/s becomes two outgoing momenta. A has components (3,4) kg·m/s. Find B’s components and explain the north–south balance.

Predict. Change one thing. Explain.

Initial total momentum is (6,0). A’s final momentum magnitude is 3 kg·m/s. Rotate A between 0° and 90° and predict B’s components. The display is a vector constraint, not a complete collision trajectory or energy model.

On narrow screens, swipe or scroll diagrams sideways to read all labels.

Explain what you noticed: Which quantity changed? Which stayed fixed? Use a momentum or impulse relationship to justify your prediction.

Show what you understand.

Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.

Original written challenge

Initial P=(10,0) kg·m/s. Final p_A=(6,2) kg·m/s. (a) Write the x balance. (b) Write the y balance. (c) Find p_B. (d) Predict how p_By changes if p_Ay increases to +3 while initial P_y remains zero.

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Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.