Aim for FivePhysics C: Mechanics

Translation between representations (TBR)

Particle with a cubic position function

  • Units 1, 2 and 4
  • 12 points
  • About 28 minutes

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

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 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 25–30 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 particle of mass 2.0 kg moves along the x-axis. Its position is given by x(t)=t3−6t2+9tx(t) = t^3 - 6t^2 + 9t, where x is in meters and t is in seconds, for 0 ≤ t ≤ 4.0 s. Let FnetF_{\text{net}} be the net force on the particle along the x-axis.

Suggested time: 28 minutes

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Part (a)

2 points

Derive expressions for the velocity v(t) and the acceleration a(t) of the particle.

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

0 / 2,500 characters

Part (b)

2 points

Describe the graph of the particle's velocity as a function of time from t = 0 to t = 4.0 s that you would draw. Include the axes and labels with units, the shape of the graph, and the values where the graph crosses the time axis, reaches its minimum, and begins and ends.

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

0 / 2,500 characters

Part (c)

2 points

Describe the graph of the net force on the particle as a function of time from t = 0 to t = 4.0 s that you would draw. Include the axes and labels with units, the shape, and the key values.

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

0 / 2,500 characters

Part (d)

2 points

Calculate the total distance traveled by the particle and its displacement from t = 0 to t = 4.0 s.

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

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Part (e)

2 points

Using your graphs from parts (b) and (c), identify every time interval during which the particle is speeding up. Explain how the graphs show this.

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

0 / 2,500 characters

Part (f)

2 points

Use your net force graph from part (c) to determine the change in the particle's velocity from t = 0 to t = 2.0 s. Show that this agrees with your expression for v(t) from part (a).

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

0 / 2,500 characters

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