2 cars a&b approach each other with same velocity 20m/s. Man in car a fires bullets at intervals of 10s.What will be time interval b/w sounds of 2 bullets heard by man in car B?

To solve the problem of the time interval between the sounds of two bullets heard by the man in car B, we need to consider the relative motion of both cars and the speed of sound. Let's break this down step by step.

Understanding the Scenario

We have two cars, A and B, moving towards each other at the same speed of 20 m/s. A man in car A fires bullets at intervals of 10 seconds. The key here is to determine how long it takes for the sound of these bullets to reach the man in car B.

Speed of Sound

The speed of sound in air is approximately 343 m/s at room temperature. This speed will be crucial in calculating how quickly the sound travels from car A to car B.

Relative Velocity

Since both cars are moving towards each other, we need to consider their relative velocity. The relative velocity of the two cars can be calculated as follows:

• Velocity of car A = 20 m/s
• Velocity of car B = 20 m/s
• Relative velocity = 20 m/s + 20 m/s = 40 m/s

Distance Between the Cars

Assuming the cars are initially a certain distance apart, let's denote this distance as D. The bullets are fired every 10 seconds, meaning the first bullet is fired at time t = 0 seconds, and the second bullet is fired at t = 10 seconds.

Time for Sound to Travel

When the first bullet is fired, the sound will take some time to reach car B. The time taken for the sound to travel from car A to car B can be calculated using the formula:

Time = Distance / Speed

However, since both cars are moving towards each other, the distance between them decreases over time. The distance when the first bullet is fired is D, and when the second bullet is fired, the distance will be less due to the movement of both cars.

Calculating the Time Intervals

Let’s denote the time taken for the sound of the first bullet to reach car B as T1 and for the second bullet as T2. The distance covered by both cars in the time taken for the sound to travel can be calculated as:

• Distance covered by car A in T1 = 20 * T1
• Distance covered by car B in T1 = 20 * T1

Thus, the effective distance for the sound to travel when the first bullet is fired is:

D - (20 * T1 + 20 * T1) = D - 40 * T1

For the first bullet, the time taken for the sound to reach car B is:

T1 = (D) / (343)

For the second bullet, the distance will be:

D - 40 * 10 = D - 400

So, the time taken for the sound of the second bullet to reach car B is:

T2 = (D - 400) / (343)

Finding the Time Interval

The time interval between the sounds of the two bullets heard by the man in car B is:

Interval = T2 - T1

Substituting the values we derived:

Interval = [(D - 400) / 343] - [D / 343]

This simplifies to:

Interval = (-400) / 343

Calculating this gives us approximately:

Interval ≈ -1.17 seconds

This negative value indicates that the second bullet sound reaches car B sooner than the first bullet sound due to the relative motion of the cars. Therefore, the time interval between the sounds of the two bullets heard by the man in car B is approximately 1.17 seconds.

Final Thoughts

This problem illustrates the fascinating interplay between motion and sound. The relative speeds of the cars significantly affect how quickly sounds travel between them, leading to interesting results like the one we just calculated. If you have any further questions or need clarification on any part of this explanation, feel free to ask!