Question

In an experiment to measure the height of a bridge by dropping a stone into the water underneath, if the error in the measurement of time is $0.1\,s$ at the end of $2\,s$, then the error in the estimation of the height of the bridge will be:
(A) $0.49\,m$
(B) $0.98\,m$
(C) $1.96\,m$
(D) $2.12\,m$

Answer 1

Hint: The height of the bridge can be determined by using the free-fall formula because the stone is dropped into the water, so the stone is said to be in free fall, and by differentiating the free fall formula with respect to time, the error in height is determined.

Formula used:
The free fall of the stone is given by,
$H = \dfrac{1}{2} \times g \times {t^2}$
Where $H$ is the height of the bridge, $g$ is the acceleration due to gravity and $t$ is the time taken by the stone to reach the water.

Complete step by step answer:
Given that,
The error in the measurement of time is, $dt = 0.1\,s$
The taken by the stone to reach the water is, $t = 2\,s$
Now,
The free fall of the stone is given by,
$H = \dfrac{1}{2} \times g \times {t^2}\,................\left( 1 \right)$
By differentiating the equation (1) with respect to the time, then the above equation is written as,
$\Rightarrow dH = \dfrac{1}{2} \times g \times 2t \times dt$
Here, $dH$ is the error in the estimation of the height.
By canceling the same terms in the numerator and in the denominator, then the above equation is written as,
$\Rightarrow dH = g \times t \times dt$
By substituting the acceleration due to gravity, the time is taken, and the error in the measurement of the time, then
\[\Rightarrow dH = 9.8 \times 2 \times 0.1\]
On multiplying the above equation, then the above equation is written as,
$\Rightarrow dH = 1.96\,m$

The error in the estimation of the height of the bridge is 1.96m. Hence, option (C) is the correct answer.

Note:
The formula of the free fall of the stone is differentiated with respect to the time because in the question it is given that the error in time measurement, so the free fall equation is differentiated with respect to the time.