---
title: "A golf ball of mass \\(m\\) is initially at rest on a horizontal tee. A golf club strikes the ball, exerting a horizontal force on it. The magnitude of the force exerted by the club on the ball as a function of time \\(t\\) is modeled by the equation \\(F(t) = C t (t_f – t)\\), where \\(C\\) is a positive constant and \\(t_f\\) is the total time the club is in contact with the ball. The collision begins at \\(t = 0\\)."
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url: "https://nerd-notes.com/ubq/117764/"
date_modified: "2026-08-04T07:56:08+00:00"
---

# A golf ball of mass \(m\) is initially at rest on a horizontal tee. A golf club strikes the ball, exerting a horizontal force on it. The magnitude of the force exerted by the club on the ball as a function of time \(t\) is modeled by the equation \(F(t) = C t (t_f – t)\), where \(C\) is a positive constant and \(t_f\) is the total time the club is in contact with the ball. The collision begins at \(t = 0\).

A golf ball of mass \(m\) is initially at rest on a horizontal tee. A golf club strikes the ball, exerting a horizontal force on it. The magnitude of the force exerted by the club on the ball as a function of time \(t\) is modeled by the equation \(F(t) = C t (t_f - t)\), where \(C\) is a positive constant and \(t_f\) is the total time the club is in contact with the ball. The collision begins at \(t = 0\).

**Part a)**  *(4 points)*

**Part b)** Using calculus, **derive** an expression for the total impulse delivered to the golf ball during the collision. Express your answer in terms of \(C\), \(t_f\), and fundamental constants. *(3 points)*

**Part c)** **Derive** an expression for the final speed \(v_f\) of the golf ball immediately after the collision. Express your answer in terms of \(m\), \(C\), \(t_f\), and fundamental constants. *(2 points)*

**Part d)** A student states that because the force curve is symmetric over time, the average force exerted on the ball during the collision is exactly half of the maximum instantaneous force exerted on the ball. **Indicate** whether the student's claim is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your answer. You may use mathematical derivations or refer to features of your graphs from part (a) in your justification. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117764/*