Question

Anjali fires two bullets from the same place at an interval of 6 minutes but Bhagwat sitting in a car approaching the place of firing hears the second fire 5 minute 32 seconds after the first firing. What is the speed of car, if the speed of sound is \(332 \mathrm{~m} / \mathrm{s} ?\) (a) \(56 \mathrm{~m} / \mathrm{s}\) (b) \(102 \mathrm{~m} / \mathrm{s}\) (c) \(28 \mathrm{~m} / \mathrm{s}\) (d) \(32 \mathrm{~m} / \mathrm{s}\)

Short answer

Answer: The speed of the car is 28 m/s.

Step-by-step solution

Convert time intervals into seconds

First, we need to convert the given time intervals into seconds for easier calculations: 6 minutes = 6 * 60 = 360 seconds 5 minutes 32 seconds = 5 * 60 + 32 = 332 seconds

Calculate distance traveled

Let's denote the distance traveled by the sound between the two firings as d1 (for the first firing) and d2 (for the second firing). Let the distance traveled by the car be x. Since the car is approaching the place of firing, the distances covered by the sound when reaching Bhagwat are (d1 - x) and (d2 - x) for the first and the second firing, respectively. We can use the formula distance = speed * time to calculate d1 and d2: d1 = 332 * 360 d2 = 332 * 332 Now, we will use the formula for the first and second firing to find the relation between x and the speed of the car (v). (d1 - x) / v = 360 (d2 - x) / v = 332

Solve for the speed of the car (v)

We will use both equations to solve for the variable v, the speed of the car. To do this, we can express x from the first equation and substitute it into the second equation: x = d1 - 360v Now, we substitute this expression into the second equation: (d2 - (d1 - 360v))/v = 332 Multiplying both sides by v: d2 - d1 + 360v = 332v Now, let's solve for v: 28v = d2 - d1 v = (d2 - d1) / 28 Finally, we can plug in the values we calculated in Step 2: v = (332 * 332 - 332 * 360) / 28 v = 28 m/s The speed of the car is 28 m/s, which corresponds to option (c) in the given choices.

Key concepts

Speed of Sound

The speed of sound is an essential concept in relative motion problems, especially when you're dealing with scenarios involving distances and time intervals. Sound travels through the air at a specific speed, which in many cases is approximately 332 meters per second. This speed can vary based on environmental conditions like temperature and pressure, but for our problem, we will consider it constant.

When sound is emitted from a source, it travels until it reaches a receiver, covering a distance determined by the time it takes to travel and the speed of sound. Knowing the speed allows us to calculate how far sound travels in any given time using the formula:

• Distance = Speed × Time

Understanding the speed of sound is crucial when calculating distances between moving objects, like Anjali firing bullets and Bhagwat in the car hearing them. By using the speed of sound, we can estimate how quickly sound travels from one point to another and analyze motion accurately.

Distance Calculation

Calculating distances in a relative motion problem involves understanding how various distances interact and use algebraic equations to find unknown values. In the context of the problem, we are dealing with two main distances: the distance the sound travels and the distance the car travels as it approaches the firing point.

Using the formula Distance = Speed × Time, we can find the distance covered by sound for each bullet fired. Let's call these distances \( d_1 \) and \( d_2 \). Here, \( d_1 = 332 \, \text{m/s} \times 360 \, \text{s} \) for the first firing and \( d_2 = 332 \, \text{m/s} \times 332 \, \text{s} \) for the second.

Besides the sound, the car is also in motion toward the source of the sound, shortening the distance that the sound waves need to travel to reach the car. If \( x \) is the distance the car travels while sound is traveling, then the equations \( (d_1 - x) \) and \( (d_2 - x) \) help set up a relationship between the sound and the car's travel. This relationship is used to solve for the speed of the car as part of finding relative motion solutions.

Time Conversion

Time conversion is a critical step in solving problems involving different time measurements. Often, in physics problems, time is provided in various units like minutes and seconds, which need to be standardized for smooth calculations. Converting these units into seconds gives a single, consistent unit to work with.

In our given problem, Anjali fires bullets 6 minutes apart, but Bhagwat hears the second shot only 5 minutes and 32 seconds after the first. By converting minutes into seconds (remember that 1 minute equals 60 seconds), we convert:

• 6 minutes into 360 seconds
• 5 minutes and 32 seconds into 332 seconds

This conversion simplifies comparison and computation, allowing us to accurately substitute these values into the formulae for distance and speed, leading to the final solution of the problem.