An airplane flies at a constant altitude H = 2000 m with…

An airplane flies at a constant altitude $H = 2000\ \mathrm{m}$ with horizontal speed $v_0 = 300\ \mathrm{m/s}$. Above point A it releases a bomb. Find (a) the time until the bomb hits the ground; (b) the horizontal distance AB it covers before landing.

Explanation of Method

How far ahead of the target to release a bomb.

Step 1: (a) Fall time

\[ t = \sqrt{\frac{2H}{g}} = \sqrt{\frac{4000}{9.81}} = \sqrt{407.7}. \]

\[t \approx 20.2\ \mathrm{s}\]

Step 2: (b) Distance AB

\[ AB = v_0 t = 300 \cdot 20.19. \]

\[AB \approx 6060\ \mathrm{m} \approx 6.1\ \mathrm{km}\]

$H = 2000\ \mathrm{m}$ $A$ $B$ $AB \approx 6060\ \mathrm{m}$ $v_0 = 300\ \mathrm{m/s}$ $t \approx 20.2\ \mathrm{s}$

Drawn to scale: in the 20.2 s it takes to fall 2 km, the bomb also moves about 6 km forward. It must be released that far before the target.

Result

\[\boxed{\,t \approx 20.2\ \mathrm{s}\,}\]

\[\boxed{\,AB \approx 6060\ \mathrm{m} \approx 6.1\ \mathrm{km}\,}\]

Remark

Note. The plane is directly above B when the bomb lands (both have the same horizontal speed). Air resistance, ignored here, would shorten AB considerably in reality.