Question

A boy is riding a bicycle on a circular track of radius 49m with a constant speed 7m/s. As a train 800m long approaches a nearby train track with a constant speed of 54km/hr. Each coach (C-1, C-2, C-3....) of train is 25m long and the train is lead by a 50 m long Engine. Boy only notices the coach nearest to him as the train passes. Tick the correct answer as seen by boy considering the initial situation as shown in figure.

• A. Engine of the train seems slowest to the boy.
• B. Coach no-6 is the fastest moving coach.
• C. Coach no-12 is the fastest moving coach.
• D. Coach no-26 is the slowest moving coach.

Answer 1

Answer: (C)

Coach no-12 is the fastest moving coach.

Answer 2

The boy is on a circular track, so his velocity is constantly changing direction. Let the boy's velocity be $\vec{v}_b$ and the train's velocity be $\vec{v}_t$. The relative velocity of the train with respect to the boy is $\vec{v}_{rel} = \vec{v}_t - \vec{v}_b$.

The boy's speed is constant at 7 m/s. Let his position on the circle be $(49 \cos \theta, 49 \sin \theta)$. His velocity is $\vec{v}_b = (-7 \sin \theta, 7 \cos \theta)$. The train's speed is 54 km/hr, which is $54 \times \frac{1000}{3600} = 15$ m/s. Assuming the train moves along the x-axis, $\vec{v}_t = (15, 0)$.

The relative velocity is $\vec{v}_{rel} = (15 - (-7 \sin \theta), 0 - 7 \cos \theta) = (15 + 7 \sin \theta, -7 \cos \theta)$. The magnitude of the relative velocity (perceived speed) is: $|\vec{v}_{rel}| = \sqrt{(15 + 7 \sin \theta)^2 + (-7 \cos \theta)^2}$ $|\vec{v}_{rel}| = \sqrt{225 + 210 \sin \theta + 49 \sin^2 \theta + 49 \cos^2 \theta}$ $|\vec{v}_{rel}| = \sqrt{225 + 210 \sin \theta + 49}$ $|\vec{v}_{rel}| = \sqrt{274 + 210 \sin \theta}$

The perceived speed varies with the boy's position on the circle ($\theta$). The maximum speed occurs when $\sin \theta = 1$, so $|\vec{v}_{rel}|_{max} = \sqrt{274 + 210} = \sqrt{484} = 22$ m/s. This happens when the boy is at the top of his circular path (assuming $\theta = \pi/2$). The minimum speed occurs when $\sin \theta = -1$, so $|\vec{v}_{rel}|_{min} = \sqrt{274 - 210} = \sqrt{64} = 8$ m/s. This happens when the boy is at the bottom of his circular path (assuming $\theta = 3\pi/2$).

The train is 800m long. The engine is 50m, and each coach is 25m. The boy observes the part of the train nearest to him. As the train passes, different coaches will be nearest to the boy at different times. Since the boy's speed is not constant in direction, the perceived speed of the train varies.

If coach no. 12 is the fastest, it means it is observed when the relative speed is maximum (22 m/s). This occurs when $\sin \theta = 1$. If coach no. 26 is the slowest, it means it is observed when the relative speed is minimum (8 m/s). This occurs when $\sin \theta = -1$.

Given the length of the train (800m) and the time it takes for the boy to move through different parts of his circle, it is plausible that coach no. 12 is observed when the boy is at the position of maximum relative speed, and coach no. 26 is observed when the boy is at the position of minimum relative speed. Therefore, coach no. 12 is the fastest moving coach from the boy's perspective.