JEE PYQ: Motion in a Straight Line - Question ID 6f21fa5c9d50 (JEE Main 2003)

In problems involving braking distances, the key physics concept is the work-energy theorem. When brakes are applied, the kinetic energy of the car is dissipated by the work done by the braking force. Assuming the braking force is constant, the work done by the brakes equals the change in kinetic energy of the car.

The relevant formulas are:

• Kinetic energy: $K = \frac{1}{2} m v^2$, where $m$ is the mass of the car and $v$ is its speed.
• Work done by braking force: $W = F \cdot d$, where $F$ is the magnitude of the braking force and $d$ is the stopping distance.

Since the car comes to rest, the work done by the brakes equals the initial kinetic energy: $F \cdot d = \frac{1}{2} m v^2$

From this, we can express the stopping distance $d$ as: $d = \frac{m v^2}{2 F}$

Notice that the stopping distance depends on the square of the speed. This is the crucial insight for solving the problem.

Step 1: Convert speeds to consistent units
The given speeds are in km/hr. To make calculations easier, convert them to m/s:

However, since we are dealing with ratios, we can avoid explicit conversion by working directly with the ratio of speeds.

Step 2: Express stopping distance in terms of speed
From the work-energy relation: $d = \frac{m v^2}{2 F}$

Let $d_1$ be the stopping distance at speed $v_1 = 50 \text{ km/hr}$, and $d_2$ be the stopping distance at speed $v_2 = 100 \text{ km/hr}$. Then:

Taking the ratio of $d_2$ to $d_1$:

Step 3: Substitute the given values
Given $v_1 = 50 \text{ km/hr}$, $v_2 = 100 \text{ km/hr}$, and $d_1 = 6 \text{ m}$:

Thus:

Conclusion: The minimum stopping distance at 100 km/hr is 24 m, which corresponds to option C.

Trap 1: Linear thinking
Many students mistakenly assume that doubling the speed doubles the stopping distance. This leads them to choose 12 m (option A). However, stopping distance depends on the square of the speed, not the speed itself.

Trap 2: Ignoring unit consistency
While the ratio method avoids explicit unit conversion, some students convert speeds incorrectly, leading to calculation errors. Always ensure units are consistent when performing numerical calculations.

Exam Tip:
When dealing with braking or stopping distances, always check if the relationship is linear or quadratic with respect to speed. The work-energy theorem is a powerful tool for such problems, and recognizing the $v^2$ dependence is key to avoiding mistakes.