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Chapter 4: Problem 499

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

Expert verified
\(400 \mathrm{~W}\)

Step by step solution

01

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: \(1 \mathrm{~g} = 0.001 \mathrm{~kg}\)
02

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 = \(20 \mathrm{~g} \times 0.001 \mathrm{~kg/g} = 0.02\mathrm{~kg}\)
03

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 = \(\frac{1}{2} \times 0.02\mathrm{~kg} \times (100\mathrm{~m/s})^2 = 100 \mathrm{~J}\)
04

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 = \(100\mathrm{~J} \times 4\mathrm{~bullets/s} = 400\mathrm{~W}\)
05

Choose the correct answer

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

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