The speed of a train T is 100 km per hour and the speed of a person P is 4 km per hour. T crosses P in 15 seconds, if P travels along the direction…

The correct option is (c) - 13.85.

Solution

To find the time taken to cross in the opposite direction, we first need to determine the length of the train (L).

Step 1: Calculate Relative Speed in the same direction
Speed of Train (V_T) = 100 km/hr
Speed of Person (V_P) = 4 km/hr
Relative Speed (Same direction) = V_T - V_P = $100 - 4 = 96$100−4=96 km/hr
Convert to m/s: $96 \times (5/18) = 80/3$96×(5/18)=80/3 m/s

Step 2: Find the Length of the Train (L)
Distance = Speed * Time
$L = (80/3) \times 15$L=(80/3)×15 seconds
$L = 80 \times 5 = 400$L=80×5=400 meters

Step 3: Calculate Relative Speed in the opposite direction
Relative Speed (Opposite direction) = V_T + V_P = $100 + 4 = 104$100+4=104 km/hr
Convert to m/s: $104 \times (5/18) = 520/18 = 260/9$104×(5/18)=520/18=260/9 m/s

Step 4: Calculate the Time taken to cross
Time = Distance / Relative Speed
Time = $400 / (260/9)$400/(260/9)
Time = $(400 \times 9) / 260$(400×9)/260
Time = $3600 / 260 = 360 / 26$3600/260=360/26
Time = $180 / 13$180/13
Time = 13.846... seconds

Rounding to two decimal places, we get 13.85 seconds.

Why the other options are incorrect

• Option (a) - 13.51: This value is too low and would result from a relative speed higher than 104 km/hr, likely due to a calculation error in the division of 3600 by 260.
• Option (b) - 13.65: This is an incorrect approximation that does not follow from the fraction 180/13.
• Option (d) - 14.05: This value is higher than the actual result; it would imply a slower relative speed than 104 km/hr, which contradicts the physics of moving in opposite directions.

Key Concept

Relative speed is the difference of speeds when moving in the same direction and the sum of speeds when moving in opposite directions.