题目

Problem

A motorcyclist and his motorcycle have a combined mass of $480\,\mathrm{kg}$ .

The motorcyclist drives down a straight road that is inclined at an angle $\theta$ to the horizontal, where $\sin\theta=\frac1{12}$ , with the engine of the motorcycle working at $3.5\,\mathrm{kW}$ . The motorcycle is moving at a constant speed of $V\,\mathrm{m\,s^{-1}}$ . The resistance to the motion of the motorcycle is modelled as a constant force with magnitude $20V$ newtons.

Find the value of $V$ .

(5)

题目中文翻译

摩托车手与摩托车的总质量为 $480\,\mathrm{kg}$ 。

摩托车手沿一条与水平面成 $\theta$ 角的直路下坡，其中 $\sin\theta=\frac1{12}$ ，发动机以 $3.5\,\mathrm{kW}$ 的功率工作。摩托车以恒定速率 $V\,\mathrm{m\,s^{-1}}$ 行驶。将摩托车所受的运动阻力视为大小为 $20V$ 牛顿的恒力。

求 $V$ 的值。

(5)

解答

解法一

思路

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摩托车匀速下坡，重力沿斜面的分量帮助运动。发动机牵引力是功率除以速率；阻力大小用未知速率 $V$ 表示为 $20V$ ，列方程时须保留这个关系。列沿斜面的平衡方程后舍去负根。

答题过程

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Let the constant speed be $V\text{ m s}^{-1}$ . The driving force is $3500/V$ N, and the component of the weight down the slope is

480g\sin\theta=480g\times\frac1{12}=40g.

Resolving down the slope,

\frac{3500}{V}+40g-20V=0.

Hence, with $g=9.8$ ,

20V^2-392V-3500=0.

The positive root is

V=\frac{392+\sqrt{392^2+4(20)(3500)}}{40}.

The other root is negative and cannot represent a speed. Therefore the speed is

\boxed{26.3\text{ m s}^{-1}}.

M2 2021 June Q1

题目

解答

解法一

思路

答题过程