Two airplanes fly one behind the other with the same speed…
Two airplanes fly one behind the other with the same speed, $d = 750\ \mathrm{m}$ apart. A shot is fired from the first (front) airplane, and the pilot of the second airplane hears it $t = 1\ \mathrm{s}$ later. What is the speed of the airplanes? The speed of sound is $c = 340\ \mathrm{m/s}$.
Explanation of Method
Hearing a shot fired from the airplane ahead.
Step 1: Sound moves through the air, not with the airplane
Once the shot is fired, the sound spreads through the air at $c = 340\ \mathrm{m/s}$ relative to the air, no matter how fast the airplane that fired it was moving. The part of the sound that reaches the second airplane travels backward, and the second airplane flies forward toward it with speed $v$.
Step 2: Closing speed
The sound and the second airplane move toward each other, so the gap between them closes at $c + v$: $d = (c + v)\,t$.
- sound (moves through the air)
- 2nd airplane
Step 3: Solve for $v$
\[ v = \frac{d}{t} - c = \frac{750\ \mathrm{m}}{1\ \mathrm{s}} - 340\ \mathrm{m/s}. \]
\[v = 410\ \mathrm{m/s} \approx 1476\ \mathrm{km/h}\]
Check from the ground
sound travels back: $ 340\ \mathrm{m/s} \cdot 1\ \mathrm{s} = 340\ \mathrm{m} $ second airplane flies forward: $ 410\ \mathrm{m/s} \cdot 1\ \mathrm{s} = 410\ \mathrm{m} $
$ 340\ \mathrm{m} + 410\ \mathrm{m} = 750\ \mathrm{m} = d\ \checkmark $
Result
\[\boxed{\,v = 410\ \mathrm{m/s} \approx 1476\ \mathrm{km/h}\,}\]
Remark
Why the original solution kept getting $-410\ \mathrm{m/s}$. It used $c - v$, which would be right if the sound had to catch up with an airplane flying away from it (a shot from the rear airplane heard in the front one). Then $750 = (340 - v) \cdot 1$ gives a negative speed, the sign that the direction was chosen wrongly. With the correct direction the answer is $+410\ \mathrm{m/s}$. The airplanes are supersonic (410 > 340 m/s), so a shot fired from the rear airplane would never reach the front one at all.