Question

A loaded bus and an unloaded bus are both moving with the same kinetic energy. The mass of the former is twice that of the later. Brakes are applied to both so as to exert equal retarding forces. If $S_1$ and $S_2$ are the distances covered by the two buses before coming to rest respectively, then:

• A. $4S_1 = S_2$
• B. $2S_1 = S_2$
• C. $S_1 = 2S_2$
• D. $S_1 = S_2$

Hint:

Stopping distance depends solely on the ratio of kinetic energy to retarding force ($\frac{K}{F}$).
Since mass does not explicitly appear in this work-energy relation when kinetic energy is kept constant, mass differences do not affect the stopping distance.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Two buses with different masses have the same kinetic energy.
If equal retarding forces are applied to stop them, we need to compare their stopping distances.

Step 2: Key Formula and Approach:
According to the Work-Energy Theorem:
The work done by the net force acting on an object is equal to the change in its kinetic energy.
\[ W = \Delta KE \] For a stopping vehicle, the work done by the retarding force $F$ over a distance $S$ reduces the initial kinetic energy $K$ to zero:
\[ F \times S = K \] \[ S = \frac{K}{F} \]

Step 3: Detailed Explanation:

• Analyze the variables:
Let the kinetic energy of the loaded bus be $K_1$ and the unloaded bus be $K_2$.
We are given $K_1 = K_2 = K$.
Let the retarding force on the loaded bus be $F_1$ and on the unloaded bus be $F_2$.
We are given $F_1 = F_2 = F$.

• Express stopping distances:
For the loaded bus:
\[ S_1 = \frac{K_1}{F_1} = \frac{K}{F} \] For the unloaded bus:
\[ S_2 = \frac{K_2}{F_2} = \frac{K}{F} \]

• Compare distances:
Since both $\frac{K}{F}$ ratios are identical, the stopping distances must be equal:
\[ S_1 = S_2 \]

Step 4: Final Answer:
The stopping distances are equal ($S_1 = S_2$), which corresponds to Option (D).