Question

Assertion (A) When two moving objects having same kinetic energy are stopped by application of equal retarding force, then both will come to rest at same distance.

Reason (R) Same amount of work is to be done to stop two moving objects.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
• B. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
• C. Assertion (A) is true but Reason (R) is false
• D. Assertion (A) is false but Reason (R) is true

Answer 1

Answer: (A)

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)

Answer 2

Explanation:

1. Analyze Assertion (A):

• According to the Work-Energy Theorem, the work done by the net force on an object is equal to the change in its kinetic energy ($W_{net} = \Delta KE$).

• When an object is stopped, its final kinetic energy ($KE_f$) becomes zero.

• The work done by the retarding force ($F$) over a stopping distance ($d$) is $W = -F \times d$ (negative sign because the force opposes motion).

• The change in kinetic energy is $\Delta KE = KE_f - KE_{initial} = 0 - KE_{initial} = -KE_{initial}$.

• Equating work done and change in kinetic energy: $-F \times d = -KE_{initial}$.

• This simplifies to $F \times d = KE_{initial}$.

• Therefore, the stopping distance $d = \frac{KE_{initial}}{F}$.

• Since both objects have the same initial kinetic energy ($KE_{initial}$) and are stopped by the same retarding force ($F$), their stopping distance ($d$) will be the same. Thus, Assertion (A) is TRUE.

2. Analyze Reason (R):

• To stop a moving object, its kinetic energy must be reduced to zero.

• The work done to achieve this change in kinetic energy is equal to the magnitude of the initial kinetic energy (as $W = \Delta KE = 0 - KE_{initial} = -KE_{initial}$, so $|W| = KE_{initial}$).

• If two objects have the same initial kinetic energy, then the same amount of work (in magnitude) must be done to stop them. Thus, Reason (R) is TRUE.

3. Check if Reason (R) explains Assertion (A):

• Reason (R) states that the same amount of work needs to be done. This work is numerically equal to the initial kinetic energy ($W = KE_{initial}$).

• From the Work-Energy Theorem, this work is also related to the retarding force and stopping distance ($W = F \times d$).

• Combining these, we get $F \times d = KE_{initial}$.

• This equation directly shows that if $KE_{initial}$ and $F$ are constant, then $d$ must also be constant. Therefore, Reason (R) provides the fundamental principle (work done equals change in kinetic energy) that explains why equal kinetic energies and equal retarding forces lead to equal stopping distances.

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).