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title: "A vehicle of mass \\(m\\) starts from rest at time \\(t = 0\\) and accelerates along a straight, horizontal track. An engine delivers an instantaneous mechanical power output \\(P(v)\\) that varies with the vehicle’s speed \\(v\\) according to the relation \\(P(v) = P_0\\left(1 – \\dfrac{v}{v_0}\\right)\\), where \\(P_0\\) and \\(v_0\\) are positive constants, for speeds \\(0 \\le v < v_0\\). Assuming resistive forces are negligible, which of the following expressions represents the time \\(t_1\\) required for the vehicle to reach a speed of \\(\\dfrac{v_0}{2}\\)?"
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url: "https://nerd-notes.com/ubq/120636/"
date_modified: "2026-08-23T04:42:26+00:00"
---

# A vehicle of mass \(m\) starts from rest at time \(t = 0\) and accelerates along a straight, horizontal track. An engine delivers an instantaneous mechanical power output \(P(v)\) that varies with the vehicle’s speed \(v\) according to the relation \(P(v) = P_0\left(1 – \dfrac{v}{v_0}\right)\), where \(P_0\) and \(v_0\) are positive constants, for speeds \(0 \le v < v_0\). Assuming resistive forces are negligible, which of the following expressions represents the time \(t_1\) required for the vehicle to reach a speed of \(\dfrac{v_0}{2}\)?

A vehicle of mass \(m\) starts from rest at time \(t = 0\) and accelerates along a straight, horizontal track. An engine delivers an instantaneous mechanical power output \(P(v)\) that varies with the vehicle's speed \(v\) according to the relation \(P(v) = P_0\left(1 - \dfrac{v}{v_0}\right)\), where \(P_0\) and \(v_0\) are positive constants, for speeds \(0 \le v < v_0\). Assuming resistive forces are negligible, which of the following expressions represents the time \(t_1\) required for the vehicle to reach a speed of \(\dfrac{v_0}{2}\)?

- **A.** \(\dfrac{m}{P_0}\int_0^{v_0} \dfrac{v}{1 - \dfrac{v}{v_0}}\,dv\)
- **B.** \(\dfrac{m}{P_0}\int_0^{v_0/2} \dfrac{v_0}{1 - \dfrac{v}{v_0}}\,dv\)
- **C.** \(\dfrac{m}{P_0}\int_0^{v_0/2} \dfrac{v}{1 - \dfrac{v}{v_0}}\,dv\)
- **D.** \(\dfrac{m}{P_0}\int_0^{v_0/2} v\left(1 - \dfrac{v}{v_0}\right)\,dv\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120636/*