A person is sitting in a train moving forward and facing the engine. He tosses a coin straight up and the coin falls behind him on the seat. From this observation, what can we conclude about the motion of the train?

Introduction / Context:
This is a classic conceptual mechanics problem about motion in non inertial frames. When a person tosses a coin inside a moving train, the path of the coin relative to the person depends on whether the train is moving uniformly or accelerating or decelerating. This question tests your understanding of relative motion and how horizontal acceleration of the train affects the apparent fall of the coin for an observer inside the train.

Given Data / Assumptions:

• The passenger is facing the engine, so forward direction is in front of him.
• He tosses the coin vertically upward relative to the train compartment.
• The coin falls behind him instead of in his hand or in front of him.
• We neglect air resistance and assume a short toss inside the compartment.

Concept / Approach:
If the train were moving with constant velocity (uniform speed in a straight line), both the passenger and the coin would share the same forward velocity. When he tosses the coin straight up, it retains this initial horizontal velocity, so it would come back to his hand. If the train accelerates forward while the coin is in the air, the train and the passenger gain extra forward speed, but the coin continues with its original forward speed. As a result, by the time the coin comes down, the passenger has moved ahead relative to the coin, so the coin appears to fall behind him. In contrast, if the train were decelerating, the coin would tend to fall in front of him.

Step-by-Step Solution:
Step 1: Initially, both the passenger and the coin share the same forward velocity of the train.Step 2: When the passenger tosses the coin straight up, he gives it no extra horizontal speed, so it retains the original horizontal velocity.Step 3: If the train maintains uniform speed, the coin and the passenger remain aligned horizontally and the coin falls back into his hand.Step 4: If the train accelerates forward during the flight of the coin, the passenger and train increase their horizontal speed.Step 5: The coin keeps its old forward speed, so relative to the passenger it lags behind.Step 6: Therefore, the observation that the coin falls behind indicates that the train is moving forward and gaining speed.

Verification / Alternative check:
Consider the opposite case of a train that is slowing down while the coin is in the air. The coin would keep its larger initial forward speed while the train slows, so it would land ahead of the passenger. This is the opposite of what is described. Another check is to imagine the same experiment in a car that suddenly accelerates; loose objects tend to slide backward relative to the car, which is consistent with the coin appearing behind the passenger in an accelerating train.

Why Other Options Are Wrong:
If the train were moving forward with uniform speed, the coin would fall back into the thrower hand, not behind. If the train were losing speed, the coin would fall ahead of him, because it would retain a higher horizontal speed than the decelerating train. The option that the train is moving backward with uniform speed does not match the given situation because both the initial conditions and the appearance of the coin do not support backward motion.

Common Pitfalls:
A common mistake is to assume that any motion of the coin away from the hand implies that the train speed is changing in an unspecified way. Another frequent error is to forget that inside a uniformly moving vehicle, simple experiments like dropping objects behave as if you were at rest. To answer correctly, focus on how extra forward acceleration of the train changes the relative position of the passenger and the coin during its flight.

Final Answer:
If the tossed coin falls behind the passenger facing forward, the train must be moving forward and gaining speed.