Question

An automobile weighing 1200 kg climbs up a hill thatrises 1 m in 20s.Neglecting frictional effects,the minimum power developed by the engine is 9000W.If g=10m/s^2,then the velocity of automobile is

Answer 1

Given data:

Mass, m = 1200 kg
Height, h = 1 m
Time, t = 20 s
Power, P = 9000 W

Solution:

Total energy, E = PE + KE
$$\begin{gathered} =mgh+\frac{1}{2}m{v}^2 \\ =m\left(gh+\frac{v^2}{2}\right) \end{gathered}$$

Power:

Power, $P=\frac{E}{t}$
$$\begin{gathered} P=\frac{m\left(gh+{\displaystyle \frac{v^2}{2}}\right)}{t} \\ gh+\frac{v^2}{2}=\frac{Pt}{m} \\ \frac{v^2}{2}=\frac{Pt}{m}-gh \\ {v}^2=2\left(\frac{Pt}{m}-gh\right) \\ v=\sqrt{2\left(\frac{Pt}{m}-gh\right)} \\ =\sqrt{2\left(\frac{9000\times 20}{1200}-10\times 1\right)} \\ =16.7m/s \end{gathered}$$

Therefore, the velocity of automobile is 16.7 m/s.