Qualitative/quantitative translation (QQT)

Crossing a river in the least time

Unit 1
8 points
About 18 minutes

You can use a calculator on this question, just like on exam day.

A shorter multipart problem that connects reasoning in words with an equation. You make and justify a claim without equations, derive an equation from a physics principle, and then explain whether the two agree or use them to predict what happens in a changed situation. On the exam: Question 4 of 4. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 15–20 minutes. This question type started in May 2025, so Physics C: Mechanics free-response questions from 2024 and earlier are built differently.

The question

A river of width d flows with a uniform current of speed u relative to the ground, parallel to the straight banks. A boat moves with constant speed $v_b$ relative to the water, where $v_b > u$ . The boat starts at point P on one bank. A student claims: “To cross the river in the least possible time, the boat should point directly across the river, even though the current will carry it downstream.”

Suggested time: 18 minutes

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Part (a)

3 points

Do you agree or disagree with the student's claim? Justify your answer without using equations.

Your answer to part (a)

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (b)

3 points

The boat is now pointed at an angle θ upstream from the line straight across the river. Derive expressions for (i) the time t for the boat to cross the river and (ii) the distance s downstream from the point directly opposite P where the boat lands. Express your answers in terms of d, u, $v_b$ , and θ, as appropriate.

Your answer to part (b)

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (c)

2 points

Explain how your expression for t from part (b) supports your reasoning in part (a). Then use your expressions to determine the angle θ at which the boat lands directly opposite P, and state whether that crossing takes more or less time than the crossing in the student's claim.

Your answer to part (c)

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

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