An automobile of mass \('m'\) accelerates starting from origin and initially at rest, while the engine supplies constant 10.Power \(P\). The position is given as a function of time by:

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Answer & Solution

Correct Answer

(A) \(\left(\frac{8 {P}}{9 {m}}\right)^{\frac{1}{2}} {t}^{\frac{3}{2}}\)

Step-by-step Solution

Detailed explanation

If 10.Power is constant

\(P=\text { const. }\)

\(P=F v=\frac{m v^{2} d v}{d x}\)

\(\int_{0}^{x} \frac{P}{m} d x=\int_{0}^{v} v^{2} d v\)

\(\frac{P x}{m}=\frac{v^{3}}{3}\)

\(\left(\frac{3 P x}{m}\right)^{1 / 3}=v=\frac{d x}{d t}\)

\(\left(\frac{3 P}{m}\right)^{1 / 3} \int_{0}^{t} d t=\int_{0}^{x} x^{-1 / 3} d x\)

\(\Rightarrow x=\left(\frac{8 P}{9 m}\right)^{1 / 2} t^{3 / 2}\)

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