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The driver of a train moving at 72 km h^...

The driver of a train moving at `72 km h^(-1)` sights another train moving at `4 ms^(-1)` on the same track and in the same direction. He instantly applies brakes to produces a retardation of `1 ms^(-2)`. The minimum distance between the trains so that no collision occurs is

A

32 m

B

64 m

C

128 m

D

256 m

Text Solution

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The correct Answer is:
To solve the problem step by step, we will follow the given information and apply the relevant physics concepts. ### Step 1: Convert the speed of the first train from km/h to m/s The speed of the first train (A) is given as 72 km/h. We need to convert this to meters per second (m/s). \[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \] Calculating this: \[ \text{Speed of A} = 72 \times \frac{5}{18} = 20 \text{ m/s} \] ### Step 2: Identify the speed of the second train The speed of the second train (B) is given as 4 m/s. ### Step 3: Calculate the relative speed of train A with respect to train B Since both trains are moving in the same direction, the relative speed of train A with respect to train B can be calculated as: \[ \text{Relative Speed (V_{AB})} = V_A - V_B = 20 \text{ m/s} - 4 \text{ m/s} = 16 \text{ m/s} \] ### Step 4: Determine the retardation of train A The retardation (deceleration) of train A is given as \(1 \text{ m/s}^2\). ### Step 5: Use the equations of motion to find the minimum distance (d) We want to find the minimum distance \(d\) such that train A comes to a stop before colliding with train B. We can use the third equation of motion: \[ V^2 = U^2 + 2a s \] Where: - \(V\) = final velocity (0 m/s, since we want A to stop) - \(U\) = initial velocity (16 m/s, relative speed) - \(a\) = acceleration (which will be negative due to retardation, so \(a = -1 \text{ m/s}^2\)) - \(s\) = distance (d) Substituting the values into the equation: \[ 0 = (16)^2 + 2(-1)d \] This simplifies to: \[ 0 = 256 - 2d \] Rearranging gives: \[ 2d = 256 \] \[ d = \frac{256}{2} = 128 \text{ m} \] ### Conclusion The minimum distance between the two trains so that no collision occurs is **128 meters**. ---
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