In an experiment to measure the velocity of water splash when a car passes over a water layer on the road, it is found that the velocity depends on weight of car, power delivered by its engine and thickness of water layer. If velocity of splash was found to be 3 m/s when a car weighing 1000 N passes over a layer of thickness 3 mm and its engine is supplying a power of 800 kW. Find the velocity (in m/s) of splash when car weighing 500 N passes over a layer of thickness 4 mm and its engine is supplying a power of just 400 kW.

Difficult

Solution

From dimensional analysis, $v = k F^{a} P^{b} t^{c}$ where t is thickness
$\begin{array}{l} \Rightarrow M^{0} L T^{- 1} = {(M L T^{- 2})}^{a} {(M L^{2} T^{- 3})}^{b} {(L)}^{c} \\ ∴ a + b = 0 \\ a + 2 b + c = 1 \\ - 2 a - 3 b = - 1 \\ ∴ a = - 1, b = 1 and c = 0 \\ ∴ \frac{v_{1}}{v_{2}} = \frac{F_{1}^{- 1} P_{1}}{F_{2}^{- 1} P_{2}} \\ \Rightarrow v_{2} = \frac{v_{1} P_{2} F_{1}}{P_{1} F_{2}} = \frac{3 \times 400 \times 1000}{800 \times 500} = 3 m / s \end{array}$

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