Mathematical routines (MR)
Launcher with a nonlinear elastic cord
- Unit 3
- 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 toy launcher uses an elastic cord to fire a 0.050 kg projectile along a horizontal, frictionless track. When the projectile is pulled back a distance x from the launch point, the cord exerts a force on it of magnitude , directed toward the launch point, where . The projectile is pulled back to x = D = 0.30 m and released from rest. It leaves the cord at the launch point, x = 0.
Suggested time: 22 minutes
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Part (a)
2 pointsDerive an expression for the work done on the projectile by the cord as the projectile moves from x = D to x = 0, in terms of C and D.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (b)
2 pointsCalculate the speed of the projectile as it leaves the cord.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
2 pointsThe cord's force is conservative. Derive an expression for the potential energy U(x) stored in the cord when the projectile is pulled back a distance x, taking U(0) = 0. Then describe the graph of U versus x for 0 ≤ x ≤ 0.30 m, including its shape and key values.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
2 pointsCalculate the power delivered to the projectile by the cord at the instant the projectile is at x = 0.15 m.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
2 pointsA student proposes replacing the cord with an ideal spring that exerts the same maximum force when pulled back to x = 0.30 m (so its spring constant is k = CD = 120 N/m). The student claims the projectile would leave the launcher faster. 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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