Question

From an automatic gun a man fires 240 bullet per minute with a speed of \(360 \mathrm{~km} / \mathrm{h}\). If each weighs \(20 \mathrm{~g}\), the power of the gun is (A) \(400 \mathrm{~W}\) (B) \(300 \mathrm{~W}\) (C) \(150 \mathrm{~W}\) (D) \(600 \mathrm{~W}\)

Short answer

\(400 \mathrm{~W}\)

Step-by-step solution

Convert the given quantities into appropriate units

First, we need to convert the given quantities into appropriate units. The speed of the bullets is given in km/h, but we need it in meters per second (m/s). To convert the speed, we can use the expression: \(1 \mathrm{~km/h} = \frac{1000 \mathrm{~m}}{3600 \mathrm{~s}}\) Further, the mass of the bullet is given in grams (g) but we need it in kilograms (kg). To convert the mass, we can use the expression: \(1 \mathrm{~g} = 0.001 \mathrm{~kg}\)

Calculate the speed in m/s and mass in kg

Now, let's convert the speed and mass of the bullet using the expressions we derived in step 1: Speed in m/s = \(360 \mathrm{ ~km/h} \times \frac{1000 \mathrm{~m}}{3600 \mathrm{~s}} = 100 \mathrm{~m/s}\) Mass in kg = \(20 \mathrm{~g} \times 0.001 \mathrm{~kg/g} = 0.02\mathrm{~kg}\)

Find the kinetic energy of one bullet

To find the power of the gun, we must first calculate the kinetic energy transferred in firing one individual bullet. The formula for kinetic energy, KE, is: KE = \(\frac{1}{2}mv^2\), where m is the mass of the bullet in kg and v is the speed of the bullet in m/s. Now, let's plug in the values of mass and speed into the formula to find the kinetic energy: KE = \(\frac{1}{2} \times 0.02\mathrm{~kg} \times (100\mathrm{~m/s})^2 = 100 \mathrm{~J}\)

Calculate the power of the gun

Now that we have the kinetic energy of one bullet, we can calculate the power of the gun. Power is the energy transferred per unit time, i.e., the rate at which bullets are fired. To find the power, multiply the kinetic energy of one bullet by the number of bullets fired per second (since the power is given in watts, which is joules per second). The gun fires 240 bullets per minute, so we can convert that to bullets per second by dividing by 60. 240 bullets per minute = \(\frac{240}{60} = 4\) bullets per second Now, we can calculate the power: Power = KE x number of bullets per second = \(100\mathrm{~J} \times 4\mathrm{~bullets/s} = 400\mathrm{~W}\)

Choose the correct answer

Finally, based on our calculations, the power of the gun is 400 W. So, the correct answer is: (A) \(400 \mathrm{~W}\)

Key concepts

Kinetic Energy Formula

Kinetic energy is a crucial concept in physics that explains the energy an object possesses due to its motion. To find the kinetic energy of an object, we use the formula:\[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the object in kilograms, and
• \( v \) is the velocity of the object in meters per second.

This formula shows that kinetic energy depends on both the mass of the object and its velocity. The faster the object moves or the heavier it is, the more kinetic energy it has. It is important to note how sensitive kinetic energy is to speed, as it is proportional to the square of the velocity. Hence, a small increase in speed results in a larger increase in kinetic energy.

Power Calculation

Power is the rate at which energy is transferred or converted. To calculate power, we use the equation:\[ P = \frac{E}{t} \]where:

• \( P \) is power measured in watts \((W)\),
• \( E \) is the energy transferred in joules \((J)\), and
• \( t \) is the time in seconds \((s)\).

In the context of firing bullets from a gun, the power calculation involves figuring out how much energy is given by each bullet and how many bullets are fired per second. This gives a practical way to measure the effectiveness or output of the gun, represented in watts. Understanding this concept allows one to not only gauge the energy efficiency of devices but also to solve various physics problems involving energy transfer.

Unit Conversions

In physics, unit conversions are essential for ensuring that calculations are accurate and comprehensible. Different units must sometimes be converted to match the standard units for a particular formula. For example:

• Converting speed from kilometers per hour \((km/h)\) to meters per second \((m/s)\): \(1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}}\)
• Converting mass from grams \((g)\) to kilograms \((kg)\): \(1 \text{ g} = 0.001 \text{ kg}\)

By properly converting units, we ensure that our calculations in physics align with the standards in formulas. This is especially important when dealing with quantities involving energy and power, where mixing units can lead to incorrect results.

Physics Problem Solving

Solving physics problems involves a systematic approach that helps one to analyze and find solutions effectively. Here are some key steps:

• **Understand the Problem**: Read the problem thoroughly to grasp what is being asked. Identify what you know and what you need to find out.
• **Identify Relevant Formulas**: Determine which physics principles and formulas apply to the given problem. Like using the kinetic energy formula or power equation for problems involving energy.
• **Convert Units if Necessary**: Make sure all measurements are in the correct units for the formulas you are using.
• **Perform Calculations**: Substitute the known values into the formulas and solve for the unknowns.
• **Check Your Work**: After obtaining a solution, review your calculations to ensure accuracy.

By following these structured steps, you build a solid foundation for tackling complex physics problems with confidence. Whether involving bullets or any other contexts, these principles hold true for effective problem-solving.