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 $$g=10$$ ms–$$2$$ 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 $$g=10$$ ms–$$2$$     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

$$T=2Lg+LV$$  $$T=L5+L300$$ $$dT=1512L$$−$$1/2dL+(1300dL)dT=125120dL+dL300=0.01dL(120+1300)=0.01dL$$[$$15300$$]$$=0.01dL=316dLL$$×$$100=316$$×$$120$$×$$100=1516$$≃$$1$$%

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