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.

Question

Infinity Learn · Accepted Answer

From dimensional analysis, $$v=kFaPbtc$$  where t is thickness⇒    $$M0LT$$−$$1=MLT$$−$$2aML2T$$−$$3bLc$$∴    $$a+b=0$$    $$a+2b+c=1$$    −$$2a$$−$$3b=$$−$$1$$∴    $$a=$$−$$1$$,$$b=1$$ and $$c=0$$∴    $$v1v2=F1$$−$$1P1F2$$−$$1P2$$⇒    $$v2=v1P2F1P1F2=3$$×$$400$$×$$1000800$$×$$500=3m/s$$

Dimensions of physical quantities

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Dimensions of physical quantities

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From dimensional analysis, v=kFaPbtc where t is thickness⇒ M0LT−1=MLT−2aML2T−3bLc∴ a+b=0 a+2b+c=1 −2a−3b=−1∴ a=−1,b=1 and c=0∴ v1v2=F1−1P1F2−1P2⇒ v2=v1P2F1P1F2=3×400×1000800×500=3m/s

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