Problem 72 - Master Resource Book in JEE Main Physics

Question

Two trains travelling on the same track are approaching each other with equal speeds of \(40 \mathrm{~ms}^{-1}\). The drivers of the trains begin to decelerate simultaneously when they are just \(2 \mathrm{~km}\) apart. If the decelerations are both uniform and equal, then the value of deceleration to barely avoid collision should be (a) \(0.8 \mathrm{~ms}^{-2}\) (b) \(2.1 \mathrm{~ms}^{-2}\) (c) \(11.0 \mathrm{~ms}^{-2}\) (d) \(13.2 \mathrm{~ms}^{-2}\)

Step-by-Step Solution

Answer

The deceleration should be \(0.8 \mathrm{~m/s^2}\).

Step 1: Understand the Problem

Two trains are approaching each other on the same track, each with an initial speed of \(40 \mathrm{~m/s}\), and they are \(2 \mathrm{~km}\) apart when deceleration starts. We need to find the shared deceleration value that prevents collision.

Step 2: Convert Units

Convert the distance from kilometers to meters: \(2 \mathrm{~km} = 2000 \mathrm{~m}\). This is necessary to keep all units consistent during calculations.

Step 3: Set Up the Equation

Use the kinematic equation: \(v^2 = u^2 + 2as\). Here, \(v = 0 \mathrm{~m/s}\) (final velocity for both trains), \(u = 40 \mathrm{~m/s}\) (initial velocity), \(s = 1000 \mathrm{~m}\) (deceleration distance for each train, as they meet in the middle), and \(a\) is the deceleration to be found.

Step 4: Rearrange and Solve the Equation

Since \(v = 0\), the equation simplifies to \(0 = (40)^2 + 2a(1000)\). Rearranging gives \(a = -\frac{(40)^2}{2 \times 1000}\). Calculating this gives \(a = -0.8 \mathrm{~m/s^2}\).

Step 5: Interpret the Result

The negative sign indicates deceleration. Therefore, the deceleration required to avoid collision is \(0.8 \mathrm{~m/s^2}\).

Key Concepts

Kinematic Equations

The kinematic equations are essential tools in physics, helping us to solve problems involving motion in a straight line with constant acceleration, like the one in the problem at hand. These equations relate the variables of motion, which include initial velocity, final velocity, acceleration, time, and displacement. This allows us to make predictions about an object's future position or velocity when we know other information. In our problem:

1. Initial velocity (\(u\)): Each train starts with a velocity of \(40 \mathrm{~m/s}\).
2. Final velocity (\(v\)): At the end of deceleration, the trains' velocities reach \(0 \mathrm{~m/s}\), indicating that they have stopped just in time to avoid a collision.
3. Displacement (\(s\)): Each train needs to decelerate over half the total distance, \(1000 \mathrm{~m}\).

By using the equation \(v^2 = u^2 + 2as\), we incorporated these values to find the deceleration (\(a\)), ensuring the trains do not collide.

Collision Avoidance

Collision avoidance is the process of preventing two moving objects from crashing into each other. In the context of our problem, it is crucial for the safety of the trains. The drivers decide to decelerate, which is to slow down systematically until they come to a halt, exactly when necessary.

To achieve collision avoidance, the trains' speed must decrease uniformly without any abrupt changes, which would indicate a non-uniform deceleration. Uniform deceleration means speed decreases at a constant rate over time, calculated as \(-0.8 \mathrm{~ms}^{-2}\) for each train.

Adopting planned strategies like this ensures there is no collision, despite the initial risk of their high speeds and shared track.

Relative Motion

Relative motion describes how the position or speed of one object is perceived in comparison to another. In the case of the two trains moving towards each other, each train views the other as moving faster than its own speed alone. This is because their speeds are additive since they are closing in.

Before deceleration, the relative speed between the trains is \(80 \mathrm{~ms^{-1}}\) - twice their individual speed. Understanding relative motion allows us to assess the urgency for collision avoidance.

When both trains decelerate simultaneously at the same rate, the relative speed begins to decrease until it is zero, meaning they have achieved collision avoidance. This careful calculation reflects the importance of relative motion concepts in preventing accidents.