Problem 9 on Motion in a Straight Line

Description:

Problem

Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 km/h in the same direction, with A ahead of B. The driver of B decides to overtake A and accelerates by 1 m/s². If after 50s, the guard of B just brushes past the driver of A, what was the original distance between them?

Solution

First, we will solve the problem as observers in Ground Frame.

Velocity of Train A, v_A = 72 km/h = 20 m/s

Total time of the event, t = 50s

Displacement of Driver of A, D = v_A × t = 20 × 50 = 1000 m

Displacement of Guard of Train B = 2L + d + D = (1800 + d) m

Acceleration of Train B, a_B = 1 m/s²

Using, S = ut + 1/2 at², for Train B, we can write,

(1800 + d) = 20 × 50 + 1/2 × 1 × (50)²

d = 450 m (Separation between two trains)

Separation between Guard (Train B) and Driver (Train A),

Separation = 400 + 450 + 400 = 1250 m

Second, we will solve the problem as observers in the frame of A’s driver.

Initial velocity of both trains w.r.t. to ground = 72 km/h = 20 m/s

Initial velocity of Train B w.r.t. to Train A, v_BA = 20 − 20 = 0 m/s

Total time of the event, t = 50 s

Displacement of Guard of B w.r.t. Driver of A = 2L + d = (800 + d) m

Acceleration of Train B w.r.t Train A, a_BA = 1 m/s²

Using, S_BA = u_BAt + 1/2a_BA t², in the frame of Train A, we can write,

(800 + d) = 0 × 50 + 1/2 × 1 × (50)²

d = 450 m(Separation between two trains)

Separation between Guard (Train B) and Driver (Train A),

Separation = 400 + 450 + 400 = 1250 m