A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is δ$$T=0.01$$ seconds and he measures the depth of the well to be $$L=20$$ meters. Take the acceleration due to gravity $$g=10ms$$−$$2$$ and the velocity of sound is $$300$$ ms−$$1$$. Then the fractional error in the measurement,δ$$L/L$$ , is closest to

Question

A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is δ$$T=0.01$$ seconds and he measures the depth of the well to be $$L=20$$ meters. Take the acceleration due to gravity $$g=10ms$$−$$2$$ and the velocity of sound is  $$300$$ ms−$$1$$. Then the fractional error in the measurement,δ$$L/L$$ , is closest to

Infinity Learn · Accepted Answer

$$2Lg+LV=T$$ After differentiating, ⇒$$2gdL2L+0=dT$$ dLV≈$$0$$⇒$$12gdLL=dT$$ After integrating, ⇒$$L=12gT2$$⇒$$T=2Lg=2$$×$$2010=2sA/SO$$,$$dLL=2dTT=2$$×$$0.012=0.01$$⇒dLL×$$100=1$$%

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