Aim for FivePhysics C: Mechanics

Mathematical routines (MR)

Bullet embedding in a block on a spring

  • Units 2, 3 and 4
  • 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 block of mass M = 0.990 kg rests on a horizontal, frictionless surface. It is attached to one end of a light horizontal spring of spring constant k = 400 N/m; the other end of the spring is attached to a wall. The spring is at its unstretched length. A bullet of mass m = 0.010 kg moving horizontally toward the wall at speed v = 300 m/s, along the spring's axis, strikes the block and embeds in it. The collision takes a very short time.

Suggested time: 22 minutes

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Part (a)

2 points

Derive an expression for the speed V of the block–bullet combination just after the collision, in terms of m, M, and v. Justify the physics principle you use.

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

0 / 2,500 characters

Part (b)

2 points

Derive an expression for the maximum compression xmax⁡x_{\max} of the spring after the collision, in terms of m, M, v, and k.

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

0 / 2,500 characters

Part (c)

2 points

Calculate the values of V and xmax⁡x_{\max}.

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

0 / 2,500 characters

Part (d)

2 points

Calculate the kinetic energy converted to other forms during the collision, and identify what forms it becomes.

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

0 / 2,500 characters

Part (e)

2 points

In a second trial, the same bullet is fired at an identical block in the same way, but this time the bullet passes all the way through the block and exits with a speed of 100 m/s. A student claims that the spring's maximum compression will be smaller in this trial than in the first. Do you agree or disagree with the student? Support your answer with a calculation.

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

0 / 2,500 characters

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