# A Body Is Moving Unidirectionally Under The Influence Of A Source Of Constant Power. Its Displacement In Time T Is Proportional To

P = C, where C is constant.

$$P = F * v = m * a * v$$ $$a = \frac{dv}{dt}$$ $$C = m * \frac{dv}{dt} * v$$

Therefore, $$C * dt = m * v * dv$$

Now taking Integrating both sides,

$$\int C * dt = \int m * v * dv$$

Therefore, $$C * t = m * \frac{v^{2}}{2}$$ $$\Rightarrow v = \frac{ds}{dt} = (\frac{2 * C * t}{3})^\frac{1}{2}$$ $$Let, {C}’ = (\frac{2 * C}{3})^\frac{1}{2}$$

where C’ is constant.

Therefore, $$ds = {C}’ * t^\frac{1}{2} * dt$$

Now taking Integrating both sides,

$$\int ds = \int {C}’ * t^\frac{1}{2} * dt$$ $$\Rightarrow s = {C}’ * \frac{2}{3} * t^\frac{3}{2}$$ $$\Rightarrow S \propto t^\frac{3}{2}$$

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