AIIMS 2005 Physics Work and Energy MCQ Question

To solve the problem, we need to analyze the motion of the block and how the retarding force affects its kinetic energy.

Given Information:

• Mass of the block, $m = 10 \, \text{kg}$
• Initial speed, $v_i = 10 \, \text{m/s}$
• Retarding force, $F = 0.1x \, \text{N}$
• Distance traveled, from $x = 20 \, \text{m}$ to $x = 30 \, \text{m}$

Step 1: Calculate Initial Kinetic Energy

The initial kinetic energy ($KE_i$) of the block can be calculated using the formula:

$$KE = \frac{1}{2} mv^2$$

Substituting the values:

$$KE_i = \frac{1}{2} \times 10 \, \text{kg} \times (10 \, \text{m/s})^2 = \frac{1}{2} \times 10 \times 100 = 500 \, \text{J}$$

Step 2: Determine Work Done by the Retarding Force

The work done ($W$) by the retarding force as the block moves from $x = 20 \, \text{m}$ to $x = 30 \, \text{m}$ can be calculated using the integral of the force over the distance traveled. The force is given as $F = 0.1x$.

The work done by the force can be expressed as:

$$W = \int_{20}^{30} F \, dx = \int_{20}^{30} 0.1x \, dx$$

Calculating the integral:

$$W = 0.1 \int_{20}^{30} x \, dx = 0.1 \left[ \frac{x^2}{2} \right]_{20}^{30}$$ $$= 0.1 \left( \frac{30^2}{2} - \frac{20^2}{2} \right) = 0.1 \left( \frac{900}{2} - \frac{400}{2} \right) = 0.1 \left( 450 - 200 \right) = 0.1 \times 250 = 25 \, \text{J}$$

Step 3: Calculate Final Kinetic Energy

The final kinetic energy ($KE_f$) can be found using the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy:

$$KE_f = KE_i + W$$

Since the retarding force does negative work, we will subtract the work done:

$$KE_f = KE_i - W = 500 \, \text{J} - 25 \, \text{J} = 475 \, \text{J}$$

Conclusion

Thus, the final kinetic energy of the block when it reaches $x = 30 \, \text{m}$ is:

$$\boxed{475 \, \text{J}}$$

Clarification of Other Options

• B) 450 joule: This option does not account for the correct calculation of work done by the retarding force, which we've found to be 25 J.
• C) 275 joule: This is significantly lower than the calculated value and suggests a misunderstanding of the work-energy principle.
• D) 250 joule: This value does not represent any stage in the energy calculations presented, indicating a miscalculation.

In conclusion, the correct answer is $A) 475 \, \text{J}$.