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Chapter 3: Problem 354

A shell of mass \(200 \mathrm{~g}\) is ejected from a gun of mass $4 \mathrm{~kg}\( by an explosion that generates \)1.05 \mathrm{KJ}$ of energy. The initial velocity of the shell is (A) \(100 \mathrm{~m} / \mathrm{s}\) (B) \(80 \mathrm{~ms}^{-1}\) (C) \(40 \mathrm{~ms}^{-1}\) (D) \(120 \mathrm{~ms}^{-1}\)

Short Answer

Expert verified
The initial velocity of the shell is (A) \(100 \mathrm{~m} / \mathrm{s}\).

Step by step solution

01

Convert the units

Before we start solving the problem, we need to make sure all the units are consistent. Convert the mass of the shell to kg and the energy generated to J: Mass of the shell: \(200~g = 200 / 1000~kg = 0.2~kg\) Energy generated by the explosion: \(1.05~KJ = 1.05 * 1000~J = 1050~J\)
02

Calculate the initial velocity of the shell

Now we can use the equation derived above and plug in the converted values for the mass of the shell and the energy generated by the explosion: \(v_{shell} = \frac{2 * E_{explosion}}{m_{shell}} = \frac{2 * 1050~J}{0.2~kg} = \frac{2100~J}{0.2~kg} = 10500~\frac{J}{kg}\)
03

Convert the velocity to the required units

Now we can convert the calculated velocity to the required units of \(\frac{m}{s}\): \(v_{shell} = 10500~\frac{J}{kg} = 10500~\frac{m^2}{s^2}\) Since velocity is the square root of the above value, the required value is: \(v_{shell} = \sqrt{10500~\frac{m^2}{s^2}} = 102.5~\frac{m}{s}\)
04

Compare the calculated value with the given options

Now we can compare the calculated initial velocity of the shell with the given options to find the correct answer. The calculated value is 102.5 m/s, which is closest to (A) \(100 \mathrm{~m} / \mathrm{s}\). Therefore, the correct answer is (A) \(100 \mathrm{~m} / \mathrm{s}\).

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