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 m s - 2 and the velocity of sound is 300 m s - 1 . Then the fractional error in the measurement,

Step-by-step solution

Total time taken, T =   2 L g + L c , c is the sound speed in air. Now, for an error δ L in L, We have an error δ T in T So, T + δ T =   2 L + δ L g + L + δ L c   =   2 L g   1 + δ L L + L c   1 + δ L L Since, δ T T is very small, hence δ L L is also small, so taking binomial approximation T + δ T =   2 L g   1 + 1 2 δ L L + L c   1 + δ L L ⇒ T + δ T = 2 L g +   2 L g   1 2 δ L L + L c