Aim for FivePhysics C: Mechanics

Translation between representations (TBR)

Block dropped on a vertical spring

  • Units 2, 3 and 7
  • 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 block of mass m = 0.50 kg is attached to the lower end of a light vertical spring of spring constant k = 20 N/m. The upper end of the spring is fixed to the ceiling. The block is held so the spring is at its unstretched length and is then released from rest. Let y be the block's distance below the release point, with downward positive. Ignore air resistance. Use g = 9.8 m/s² (answers that use g = 10 m/s² are also accepted).

Suggested time: 28 minutes

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Part (a)

2 points

Describe the free-body diagram of the block at its lowest point. List each force, name the object that exerts it, give its direction, and compare the lengths of the arrows.

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

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Part (b)

2 points

Using conservation of energy, derive an expression for the maximum distance ymax⁡y_{\max} the block falls below the release point, in terms of m, k, and g. Then calculate its value.

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

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Part (c)

3 points

Taking the gravitational potential energy to be zero at the release point, describe graphs of the spring potential energy UsU_s, the gravitational potential energy UgU_g, and the block's kinetic energy K as functions of y, from y = 0 to y=ymax⁡y = y_{\max}, all on the same axes. Include the shape of each graph and its key values.

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 maximum speed of the block in terms of m, k, and g. Then calculate its value.

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

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Part (e)

2 points

Explain how the shape of your kinetic energy graph from part (c) is consistent with the free-body diagram you described in part (a) and with the location of maximum speed in part (d).

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Part (f)

1 point

Calculate the time it takes the block to travel from the release point to its lowest point for the first time.

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

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