---
title: "A boat of mass \\(m\\) glides across the surface of a calm lake with an initial speed \\(v_0\\). At time \\(t = 0\\), the engine is shut off, and the water exerts a net resistive force of magnitude \\(F_R = bv + cv^2\\) on the boat, where \\(b\\) and \\(c\\) are positive constants and \\(v\\) is the instantaneous speed. Which of the following integral expressions correctly gives the elapsed time \\(T\\) required for the boat’s speed to decrease to \\(\\dfrac{1}{2}v_0\\)?"
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url: "https://nerd-notes.com/ubq/124121/"
date_modified: "2026-09-28T13:30:31+00:00"
---

# A boat of mass \(m\) glides across the surface of a calm lake with an initial speed \(v_0\). At time \(t = 0\), the engine is shut off, and the water exerts a net resistive force of magnitude \(F_R = bv + cv^2\) on the boat, where \(b\) and \(c\) are positive constants and \(v\) is the instantaneous speed. Which of the following integral expressions correctly gives the elapsed time \(T\) required for the boat’s speed to decrease to \(\dfrac{1}{2}v_0\)?

A boat of mass \(m\) glides across the surface of a calm lake with an initial speed \(v_0\). At time \(t = 0\), the engine is shut off, and the water exerts a net resistive force of magnitude \(F_R = bv + cv^2\) on the boat, where \(b\) and \(c\) are positive constants and \(v\) is the instantaneous speed. Which of the following integral expressions correctly gives the elapsed time \(T\) required for the boat's speed to decrease to \(\dfrac{1}{2}v_0\)?

- **A.** \(\displaystyle \int_{0}^{v_0/2} \dfrac{m}{bv + cv^2}\,dv\)
- **B.** \(\displaystyle \int_{v_0}^{v_0/2} \dfrac{m}{bv + cv^2}\,dv\)
- **C.** \(\displaystyle -\int_{v_0/2}^{v_0} \dfrac{m}{bv + cv^2}\,dv\)
- **D.** \(\displaystyle \int_{v_0/2}^{v_0} \dfrac{m}{bv + cv^2}\,dv\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124121/*