Translation between representations (TBR)

Cart bouncing off a force sensor

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. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator allowed. The CED suggests 25–30 minutes.

The question and its sources

A cart of mass 0.50 kg moves to the right at 0.80 m/s on a level, low-friction track toward a force sensor with a spring bumper fixed at the end of the track. Take right as positive. The cart touches the bumper at t = 0.10 s and leaves it at t = 0.30 s, moving to the left at 0.60 m/s.

Force sensor graph

The force sensor records the horizontal force the bumper exerts on the cart. The magnitude of the force increases linearly from 0 at t = 0.10 s to a maximum value F_max at t = 0.20 s, then decreases linearly back to 0 at t = 0.30 s. The graph of force magnitude versus time is a symmetric triangle.

Source: Hypothetical lab setup

Suggested time: 28 minutes

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Part (a)

1 point

Describe the free-body diagram of the cart at t = 0.20 s, while it is pressing against the bumper. List each force, name the object that exerts it, give its direction, and compare the lengths of the arrows.

Your answer to part (a)

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Part (b)

4 points

Let v_i be the cart's initial speed, v_f its final speed (both magnitudes), and Δt the contact time. (i) Starting from the impulse–momentum theorem, derive an expression for F_max in terms of m, v_i, v_f, and Δt. (ii) Calculate the impulse exerted on the cart by the bumper (magnitude and direction) and F_max.

Your answer to part (b)

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Part (c)

3 points

Describe the graph of the cart's velocity versus time from t = 0 to t = 0.40 s. Include values at t = 0.10 s, 0.20 s, and 0.30 s, and describe how the slope changes.

Your answer to part (c)

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Part (d)

2 points

Explain how your velocity graph is consistent with the force graph described in the stimulus.

Your answer to part (d)

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Part (e)

2 points

The bumper is replaced with a stiffer one. The cart has the same initial and final velocities, but contact now lasts only 0.10 s, and the force graph is still a symmetric triangle. Describe how the force–time graph changes, and calculate the new maximum force.

Your answer to part (e)

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