AS 91524 · 2025 paper · momentum & circular motion

Question 1(b) Merit

Titan is the largest of Saturn's moons. Ignore every other body, and assume Titan is in a perfectly circular orbit.

M Saturn	5.68×10²⁶ kg
r Saturn	5.82×10⁷ m
M Titan	1.35×10²³ kg
centre-to-centre	1.22×10⁹ m

Two ice particles around Saturn collide and stick together without losing mass. A (2.45 kg) moves at 8.00 m s⁻¹ to the right on the diagram; B (3.20 kg) moves at 15.0 m s⁻¹ downwards, at right angles to A.

Calculate the size and direction of their combined momentum, relative to A's initial velocity.

The 2025 paper, page 3 ↗ Marking schedule, page 1 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1 What is each particle’s momentum before the collision, and which way does it point?
p_A = 2.45 × 8.00 = 19.6 kg m s⁻¹, pointing right on the diagram. p_B = 3.20 × 15.0 = 48.0 kg m s⁻¹, pointing down: at right angles to A’s.
2 Momentum is conserved when they stick. How do you add two momenta that are at right angles?
As vectors, nose to tail. They make a right-angled triangle with the total as the hypotenuse: p = √(19.6² + 48.0²) = √2688.16 = 51.85 kg m s⁻¹.
3 The question wants the direction relative to A’s velocity. Which two sides of the triangle give that angle?
B’s momentum is opposite the angle and A’s is next to it: tan θ = 48.0 / 19.6, so θ = 67.79°.

Answer 51.8 kg m s⁻¹, at 67.8° to A’s original velocity, turned towards the way B was moving.

What the marker wanted

Achieved One of: both momenta right (19.6 and 48.0 kg m s⁻¹); a labelled vector diagram with the arrows the right way round; the angle on its own; or 67.6 kg m s⁻¹ from adding the two sizes as plain numbers.
Merit Both halves: the size, 51.8 kg m s⁻¹, and the angle measured from A’s direction, 67.8°.

The schedule’s worked answer calls the direction 67.8° “anticlockwise” from A. On the paper’s diagram A moves right and B moves down, so the total points below A’s line, which is clockwise. The Merit column asks only for the size and the angle from A.

In our words, from NZQA’s 2025 marking schedule.

Common mistake

Adding 19.6 and 48.0 to get 67.6 kg m s⁻¹. The two momenta are at right angles, so they add like the sides of a triangle, by Pythagoras. NZQA’s 2025 report names adding vector sizes as plain numbers as a common reason for Not Achieved, and the schedule gives 67.6 kg m s⁻¹ no more than Achieved.

How did it go?

Got it moves it further out. Review brings it back tomorrow.