A helicopter hovers at a height H = 980 m . A bomb is…

A helicopter hovers at a height $H = 980\ \mathrm{m}$. A bomb is released from it and falls freely. After what time does the pilot hear the explosion? The speed of sound is $c = 340\ \mathrm{m/s}$.

Explanation of Method

When does the pilot hear the bomb explode?

Step 1: Fall time

\[ t_f = \sqrt{\frac{2H}{g}} = \sqrt{\frac{1960}{9.81}} \approx 14.13\ \mathrm{s}. \]

Step 2: Time for the sound to come back

\[ t_s = \frac{H}{c} = \frac{980}{340} \approx 2.88\ \mathrm{s}. \]

Step 3: Total

\[ t = t_f + t_s = 14.13 + 2.88. \]

\[t \approx 17.0\ \mathrm{s}\]

$H = 980\ \mathrm{m}$ $\text{bomb}$ $\text{sound}$ $\text{fall: } 14.13\ \mathrm{s}$ $\text{sound}$ $2.88$ $0$ $14.13$ $17.02\ \mathrm{s}$

Left: the bomb falls 980 m, then the sound of the explosion travels the same 980 m back up. Right: the two times on a timeline, to scale.

Result

\[\boxed{\,t \approx 17.0\ \mathrm{s}\,}\]

Remark

Note. The helicopter hovers, so the sound travels exactly the height $H$. Compare with Gravitational Kinematics 003, where the total time is known and the height is the unknown.