AS 91523 · 2025 paper · Doppler effect

Question 1(c) Merit to Excellence

Martyn and Kerryn are riding jet skis on a lake. Martyn's jet ski has an air horn that emits 1043 Hz. Kerryn stops, and Martyn travels directly towards her at 27.3 m s⁻¹. The speed of sound in air is 343 m s⁻¹.

f source	1043 Hz
v source	27.3 m s⁻¹
v sound (air)	343 m s⁻¹

Kerryn floats very close to the jet ski's path, while her friend Erin watches from the bank. Both are level with point B, where the jet ski passes closest to them; Erin is further from the path. A graph on the paper shows the frequency each of them hears as the jet ski travels from A, through B, to C.

Explain why the frequency Kerryn hears is the same as, or different from, the frequency Erin hears.

You need the paper’s diagram for this part. It is a graph of the frequency Kerryn and Erin each hear as the jet ski travels from A to C. The map of where they are is on page 2. The 2025 paper, page 3 ↗

Marking schedule, page 2 ↗

Work it out

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

1 Read the graph first. Where do the two lines agree, and where do they differ?
Both start at the same high frequency, about 1130 Hz, and both finish at the same low one, about 970 Hz. In between they differ: Kerryn’s stays high until B and then drops almost straight down, while Erin’s starts falling well before B and finishes well after it.
2 Far away at A, which way is the jet ski moving compared with the line to each of them?
Almost straight towards both of them: from that far off, the lines to Kerryn and to Erin both point nearly along the path. So both hear the full approaching shift, 1043 × 343 / (343 − 27.3) = 1133 Hz, the level both lines start at.
3 Only the part of the jet ski’s velocity along the line to a listener shifts the frequency they hear. What happens to that part for Kerryn as the jet ski reaches B?
Kerryn is almost on the path, so the line to her points along the path right up until the jet ski is level with her. All 27.3 m s⁻¹ counts as approaching until B, and all of it counts as receding just after. Her frequency holds at 1133 Hz, then drops suddenly at B to 966 Hz.
4 Now Erin, who is further from the path. How does the line to her change as the jet ski goes by?
It swings round steadily, making a bigger and bigger angle with the path. The part of the velocity towards her shrinks smoothly, so her frequency falls gradually. At B the jet ski moves straight across her line of sight, neither towards her nor away, so there is no shift: she hears the horn’s own 1043 Hz. After B the part moving away from her grows smoothly, and her frequency keeps falling gradually.
5 And far away at C?
The jet ski is moving almost straight away from both of them, so both hear the full receding shift, 1043 × 343 / (343 + 27.3) = 966 Hz: the level both lines end at.

Answer The same far away at A and at C, different near B. At A the jet ski is heading almost straight at both of them, so both hear about 1133 Hz; at C it is heading almost straight away from both, so both hear about 966 Hz. Near B only the part of its velocity along each person’s line of sight counts. Kerryn is almost on the path, so that part stays at the full speed until B and then reverses: her frequency drops suddenly. Erin is further away, so the part towards her shrinks gradually to zero at B, where she hears the horn’s own 1043 Hz, and the part away from her then grows gradually: her frequency falls smoothly.

What the marker wanted

Achieved Explains A or C: far away, the jet ski is moving directly towards (or away from) both of them, so they hear the same frequency.
Merit Explains B, where the jet ski moves across their line of sight with no part of its velocity towards either of them, so there is no shift and both hear 1043 Hz; or explains Erin’s gradual change as the part of the velocity along her line of sight changes.
Excellence The whole picture: A, the change around B for both of them, and C, each with its reason.

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

Common mistake

Explaining the difference by distance alone: Erin hears something different because she is further away. NZQA’s 2025 report lists this among answers that did not reach Achieved. Distance changes how loud it is; the shift depends on the part of the velocity along the line to the listener.

How did it go?

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