Two Rifle sound Problem

Question

Two rifles are fired from the same place at a difference of 11 minutes 45 seconds. But a man who is coming towards the same place in a train hears the second sound after 11 minutes. Find the speed of the train. (Assuming speed of the sound = 330 m/s).

Solution provided in my notebook : Distance covered by man in 11 minutes = Distance covered by sound in 45 seconds.

I can't figure it out why the distance covered by the man in 11 min, and the sound in 45 seconds should be same. Also I can't understand why sound travels for only 45 seconds.

Please help me. Make me understand, the explanation behind the solution given in my notebook.

Answer 1

Think of it like this: imagine the man heard the first sound at a certain moment at position $x$. While the man travels by train, imagine a second stationary man at this position. If the train was stationary, then both men would hear the second sound exactly after $11$ minutes and $45$ seconds. Clearly, the stationary man hears it after $11$ minutes and $45$ seconds, but the man in the problem heard it only after $11$ minutes. Why? Because he traveled some distance towards the source of the sound in this $11$ minutes. Now think of this distance: the sound would take exactly $45$ seconds to travel it, because the two men hears it at a difference of $45$ seconds and this distance is the distance between the two men. Again, the man in the train traveled this distance in $11$ minutes by train. So we see that the distance covered by the train in $11$ minutes is exactly the distance covered by the sound in $45$ seconds, which is exactly the distance between the two men after $11$ minutes.

Answer 2

Let's draw the distance axis. Choose point $O$, the origin of the sound. When the traveler hears the first sound, he is at point $A$. When he hears the second sound, he is at point $B$, somewhere closer to origin. Let's also assume that the first shot is at time $t=0$.

The time when the sound arrives at point $A$ is given by $c=\frac{OA}{t_A}$, so $t_A=\frac{OA}{c}$. Here $c$ is the speed of sound $c=330m/s$.

The time the sound from the second shot arrives at point $B$ is $t_B=\Delta t+\frac{OB}{c}$, where $\Delta t=11$ minutes and $45$ seconds is the time between the two shots.

You are given that $t_B-t_A=11$ minutes. If you plug the two expressions for $t_A,t_B$ into this equation, you get $$45s+\frac{OB}{c}-\frac{OA}{c}=0$$ So during these $45$ seconds, the sound (speed $c$) traveled the distance $BA$. At the same time, the train with velocity $v$ traveled the same distance in $11$ minutes (it was h=given to you in the problem).