JEE PYQ: Motion in a Straight Line - Question ID d3f8ccfcdf44 (JEE Main 2024)

Core Formula & Concept:

When two objects move towards each other under constant retardation (negative acceleration), we analyze each object separately using the kinematic equations for uniformly accelerated motion. The key formulas are:

• Final velocity squared: $v^2 = u^2 + 2 a s$
• Displacement: $s = u t + \frac{1}{2} a t^2$
• Time to stop: $t = \frac{v - u}{a}$ (when final velocity $v = 0$)

Here, both cars start with initial speed $u = 20 \, \text{m/s}$, retard at $a = -2 \, \text{m/s}^2$, and we need the distance each car travels before stopping. The initial separation is $300 \, \text{m}$, and the final separation is the initial separation minus the sum of the distances traveled by both cars.

Step-by-Step Derivation:

Step 1: Identify the stopping distance for one car

For one car, initial velocity $u = 20 \, \text{m/s}$, final velocity $v = 0$, and acceleration $a = -2 \, \text{m/s}^2$. Using the equation: $v^2 = u^2 + 2 a s$ Substitute $v = 0$: $0 = (20)^2 + 2(-2) s \implies 0 = 400 - 4 s \implies 4 s = 400 \implies s = 100 \, \text{m}$ So, each car travels $100 \, \text{m}$ before coming to rest.

Step 2: Calculate the total distance covered by both cars

Since both cars are identical and apply brakes simultaneously, each travels $100 \, \text{m}$. Therefore, the total distance covered by both cars is: $100 \, \text{m} + 100 \, \text{m} = 200 \, \text{m}$

Step 3: Determine the final separation

The initial separation between the cars is $300 \, \text{m}$. After both cars stop, the distance between them is: $300 \, \text{m} - 200 \, \text{m} = 100 \, \text{m}$

Conclusion:

The distance between the two cars when they come to rest is $100 \, \text{m}$. Thus, the correct answer is Option B.

Common Traps & Exam Tip:

Trap 1: Incorrect sign for acceleration. Students often confuse the sign of acceleration. Since the cars are retarding, acceleration must be negative ($a = -2 \, \text{m/s}^2$). Using a positive value leads to incorrect stopping distance.

Trap 2: Adding velocities instead of distances. Some students mistakenly add the velocities of the two cars ($20 + 20 = 40 \, \text{m/s}$) and use this in kinematic equations, which is incorrect. The problem requires analyzing each car's motion separately.

Trap 3: Forgetting that both cars move. A common oversight is calculating the stopping distance for only one car and subtracting it from $300 \, \text{m}$. Both cars contribute to closing the gap.

Exam Tip: Always draw a diagram to visualize the scenario. Label initial positions, velocities, and accelerations. This helps in avoiding sign errors and ensures both objects are accounted for.