Aim for FivePhysics C: Mechanics

Mathematical routines (MR)

Motorboat coasting against linear drag

  • Units 1 and 2
  • 10 points
  • About 22 minutes

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

A multipart problem where you use math, including calculus, to analyze a physical situation. You draw or sketch a representation such as a free-body diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and then make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 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 20–25 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 motorboat of mass m = 400 kg moves in a straight line across a calm lake. At time t = 0, when the boat's speed is v0=8.0v_0 = 8.0 m/s, the engine is shut off. After that, the only horizontal force on the boat is a resistive force from the water, F⃗=−kv⃗\vec{F} = -k\vec{v}, where k = 100 kg/s and v⃗\vec{v} is the boat's velocity. Treat the boat as a particle.

Suggested time: 22 minutes

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Part (a)

2 points

Describe the free-body diagram of the boat at a time after the engine is shut off. List every force, name the object that exerts it, and give its direction. Represent each force as a single force (no components), and compare the lengths of any arrows that should be equal.

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

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Part (b)

3 points

Starting with Newton's second law, derive an expression for the speed v of the boat as a function of time t after the engine is shut off. Express your answer in terms of v0v_0, k, m, t, and physical constants, as appropriate.

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

0 / 2,500 characters

Part (c)

1 point

Calculate the time after the engine is shut off when the speed of the boat is 2.0 m/s.

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

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Part (d)

2 points

Derive an expression for the position x of the boat as a function of time, taking x = 0 at t = 0. Then calculate the total distance the boat travels after the engine is shut off, if it is allowed to coast forever.

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

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Part (e)

2 points

A second boat has twice the mass of the first boat but the same shape, so it experiences the same constant k. It also coasts from 8.0 m/s after its engine is shut off. A student claims that the heavier boat coasts the same total distance, because the heavier boat has more momentum but the drag force on it is the same. Do you agree or disagree with the student? Justify your answer.

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

0 / 2,500 characters

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