Problem 354 - JEE Physics

Question

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) \(120 \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

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

Step 1: 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: \(1.05~KJ = 1.05 * 1000~J = 1050~J\)

Step 2: 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: \(v_{shell} = \frac{2 * E_{explosion}}{m_{shell}} = \frac{2 * 1050~J}{0.2~kg} = \frac{2100~J}{0.2~kg} = 10500~\frac{J}{kg}\)

Step 3: 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: \(v_{shell} = \sqrt{10500~\frac{m^2}{s^2}} = 102.5~\frac{m}{s}\)

Step 4: 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) \(100 \mathrm{~m} / \mathrm{s}\).

Key Concepts

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. In this problem, the energy from an explosion is used to propel a shell. The explosion releases a significant amount of kinetic energy, quantified in the problem as 1050 Joules. This energy is transferred to the shell, giving it a high speed as it is ejected from the gun.

To calculate kinetic energy, you can use the formula:\[ KE = \frac{1}{2}mv^2 \]where \(KE\) is the kinetic energy, \(m\) is the mass of the object, and \(v\) is its velocity. However, in this exercise, we manipulate this formula to solve for the velocity given the energy and mass, using:\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]This formula helps find the velocity the shell achieves when all the energy from the explosion is converted to kinetic energy of the shell. This understanding of kinetic energy and how it transforms into motion is key to grasping the solution to this problem.

Unit Conversion

Unit conversion is pivotal when solving physics problems to ensure consistency and accuracy. Different units measure the same physical quantity but can vary greatly in magnitude. In the context of this exercise, we begin with a mass given in grams (g) and energy in kilojoules (kJ), needing conversion to kilograms (kg) and joules (J) respectively to align with standard units in physics.

To convert the mass:

• We change 200 grams into kilograms by dividing by 1000:
• \(200 \text{ g} = \frac{200}{1000} \text{ kg} = 0.2 \text{ kg} \)

And for the energy:

• We convert 1.05 kJ to J by multiplying by 1000:
• \(1.05 \text{ kJ} = 1.05 \times 1000 \text{ J} = 1050 \text{ J}\)

Consistent units ensure that our calculations remain correct, especially when using formulas that integrate multiple different units of measure such as mass and energy.

Physics Problem Solving

Solving physics problems often requires a systematic approach to apply theoretical concepts to real-world scenarios. Here's an effective method to tackle physics problems like the one involving the shell and gun:

***Problem Analysis:***

• Identify what is given in the problem (mass, energy) and what is required (initial velocity).
• Understand the underlying physics concepts such as conservation of momentum or kinetic energy involved.

***Step-by-Step Solution:***

• Convert all measurements to consistent units, as discussed in unit conversion.
• Use formulas that connect what you know to what you need to find.
• Substitute known values into these formulas to solve for unknowns.

***Verification:***

• After calculation, ensure results make sense physically. Compare with options provided.
• Use part of the verification process to refine understanding and check for possible calculation errors.

Taking this structured approach ensures clarity and reduces errors, making problem solving in physics clearer and more manageable.