An explosion takes place at a certain distance from an army camp. As soon as the sensor in the camp receives the sound of the explosion, a drone starts flying towards the spot of explosion. The drone clicks a picture from the spot and the camp receives it at the same time. Immediately another drone starts flying to the spot and it also sends a picture as soon as it reaches the spot. The two pictures were received at 5:02 PM and 5:05 PM, respectively. If the speed of the drones is 30 m/s, at what time did the explosion take place? Assume that the speed of sound is 300 m/s.

The correct option is (c) - 4:58:42 PM.

[as per provisional answerkey]

Solution

Let the distance between the army camp and the explosion spot be D meters.
Speed of sound (V_s) = 300 m/s.
Speed of drone (V_d) = 30 m/s.

Step 1: Analyze the first drone's timeline.
1. The sound travels from the spot to the camp. Time taken = $D/300$.
2. As soon as the sound is received, the drone starts flying to the spot. Time taken = $D/30$.
3. The picture is sent back (assuming electronic transmission is instantaneous as per the "receives it at the same time" phrasing for the first drone).
Total time from explosion to 1st picture receipt = $(D/300)+(D/30)$.
This time corresponds to 5:02 PM.

Step 2: Analyze the second drone's timeline.
1. The second drone starts exactly when the first picture is received (5:02 PM).
2. It travels the same distance D. Time taken = $D/30$.
3. It reaches the spot and sends the picture at 5:05 PM.
Time elapsed between 5:02 PM and 5:05 PM = 3 minutes = 180 seconds.
Therefore, $D/30=180$ seconds.
$D=180\times 30=5400$ meters.

Step 3: Calculate the time of explosion.
We know the first picture was received at 5:02 PM.
The total time elapsed from the explosion to the first picture receipt is:
Time = $(D/300)+(D/30)$
Time = $(5400/300)+(5400/30)$
Time = 18 seconds + 180 seconds = 198 seconds.

Step 4: Subtract the elapsed time from the receipt time.
Explosion time = 5:02:00 PM - 198 seconds.
198 seconds = 3 minutes and 18 seconds.
5:02:00 PM - 3 minutes = 4:59:00 PM.
4:59:00 PM - 18 seconds = 4:58:42 PM.

Why the other options are incorrect

• Option (a) - 4:59:00 PM: This result is obtained if one only subtracts the drone's travel time (180 seconds) from the first receipt time, neglecting the time taken for the sound to reach the camp initially.
• Option (b) - 4:59:02 PM: This is a calculation error likely arising from incorrectly subtracting 178 seconds instead of 198 seconds, or misinterpreting the 3-minute interval.
• Option (d) - 4:56:32 PM: This value is significantly earlier than the actual time and would imply a much larger distance or a slower speed of sound/drone than provided in the data.

Key Concept

Relative time intervals in kinematics: The total time from an event to a signal receipt is the sum of the travel times of all sequential carriers (sound followed by the drone).