Problem 6 - Master Resource Book in JEE Main Physics

Question

Two cars \(A\) and \(B\) are travelling in the same direction with velocities \(v_{A}\) and \(v_{B}\left(v_{A}>v_{B}\right) .\) When the car \(A\) is at a distance \(s\) behind car \(B\), the driver of the car \(A\) applies the brakes producing a uniform retardation a, there will be no collision when (a) \(s<\frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\) (b) \(s=\frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\) (c) \(s \geq \frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\) (d) \(s \leq \frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\)

Step-by-Step Solution

Answer

(c) \(s \geq \frac{(v_A - v_B)^2}{2a}\)

Step 1: Understand the Problem

We need to determine the condition under which car A will not collide with car B, given that car A is initially at a distance \(s\) behind car B and is applying brakes with constant retardation \(a\).

Step 2: Analyze Equations of Motion

Car A is moving with an initial velocity \(v_A\) and starts braking, decelerating uniformly. The stopping distance \(d\) needed for car A to come to rest is given by the equation of motion: \(d = \frac{v_A^2}{2a}\). However, since car B is also moving with velocity \(v_B\), we need to consider the relative velocity \((v_A - v_B)\) between the two cars.

Step 3: Calculate Relative Stopping Distance

To avoid collision, car A must stop before reaching car B. This can be formulated as: \(d_{relative} = \frac{(v_A - v_B)^2}{2a}\), representing the stopping distance required for car A relative to car B.

Step 4: Compare Distance \(s\) to Relative Stopping Distance

To avoid collision, the distance \(s\) should be at least equal to or greater than \(d_{relative}\). This is expressed as: \(s \geq \frac{(v_A - v_B)^2}{2a}\).

Step 5: Validate Solution Against Options

Review the multiple-choice options given: - (a) \(s<\frac{(v_A - v_B)^2}{2a}\) - (b) \(s=\frac{(v_A - v_B)^2}{2a}\) - (c) \(s \geq \frac{(v_A - v_B)^2}{2a}\) - (d) \(s \leq \frac{(v_A - v_B)^2}{2a}\). The correct condition for no collision is \(s \geq \frac{(v_A - v_B)^2}{2a}\), which corresponds to option (c).

Key Concepts

Uniform Retardation

Uniform retardation, also known as uniform deceleration, is a concept from physics that describes consistent decrease in velocity over time, applied in contexts such as vehicles stopping. Imagine driving your car and then pressing the brakes; the car slows down uniformly until it halts. This change in velocity, happening at a constant rate, is precisely what we call uniform retardation.

When a vehicle experiences uniform retardation, its velocity decreases at a steady rate, which can be particularly crucial when calculating stopping distances in motion-related problems. It is similar to the reverse of uniform acceleration but involves slowing down. The value of this retardation is represented by the parameter 'a' in physics equations.

• This parameter helps in determining how quickly a vehicle can come to a stop, especially when it initially has high speeds.
• A higher retardation value means the vehicle slows down faster.

Understanding uniform retardation is essential when analyzing or predicting the stopping behavior of a vehicle under braking conditions.

Stopping Distance

Stopping distance is a vital concept in understanding motion, especially in vehicle safety. It is the distance a vehicle travels before it completely stops, after the brakes are applied. Calculating stopping distance involves initial speed and the rate of deceleration (or retardation).

The formula for calculating stopping distance is \[d = \frac{v^2}{2a}\], where:

• \(d\) is the stopping distance,
• \(v\) is the initial velocity,
• and \(a\) is the uniform retardation.

However, in scenarios involving two objects, like cars, moving in the same direction, we need to consider the relative stopping distance. This is because if both cars are moving, the distance between them changes as each decelerates differently.

Relative stopping distance can be understood as how much ground one car must cover before stopping, considering the motion of the other car. For example, if car A is initially behind car B, car A's relative stopping distance is calculated using \[d_{relative} = \frac{(v_A - v_B)^2}{2a}\], which allows us to predict if car A will stop before reaching car B and hence avoid collision.

Equations of Motion

Equations of motion are integral to solving problems involving moving objects. They provide a framework for understanding how an object moves in terms of time, velocity, acceleration, and distance. These equations describe how these variables interrelate, making it easier to predict future motion.

For uniform acceleration or retardation scenarios, there are three main equations frequently used:

• The equation for final velocity: \[v = u + at\], where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time.
• The equation for distance or displacement: \[s = ut + \frac{1}{2}at^2\], describing how far an object travels in time \(t\) given an initial velocity \(u\) and acceleration \(a\).
• The equation relating velocity with displacement: \[v^2 = u^2 + 2as\], connecting the change in speed with the distance covered.

In the car scenario from the exercise, the third equation becomes useful in determining stopping distances. By understanding how each of these equations applies, we get a complete picture of how vehicles or any objects in motion behave under constant acceleration or retardation. This knowledge is pivotal in predicting outcomes like whether a collision will occur based on given conditions.