A boy sitting in a train moving with uniform velocity drops a coin out of the window. To a man standing on the ground, what path does the coin appear to follow?

Introduction / Context:
This question illustrates the concept of relative motion and projectile motion in Newtonian mechanics. Objects dropped from moving vehicles do not simply fall straight down when observed from the ground frame. Understanding the path of the coin helps clarify how horizontal velocity and vertical acceleration combine to produce a projectile trajectory.

Given Data / Assumptions:
- The train is moving with a uniform (constant) horizontal velocity.- The boy is inside the train and drops a coin out of the window.- We ignore air resistance for simplicity and assume uniform gravitational acceleration downward.- The observer is a man standing on the ground watching the coin.

Concept / Approach:
At the instant the coin is released, it has the same horizontal velocity as the train. After release, there is no significant horizontal force, so it continues to move horizontally with that constant velocity. At the same time, gravity accelerates the coin downward. The combination of constant horizontal velocity and increasing downward velocity due to gravity forms a projectile motion, which mathematically describes a parabolic path when viewed from the ground frame.

Step-by-Step Solution:
1. Let the uniform horizontal velocity of the train be u in the direction of motion.2. When the boy drops the coin, it already has horizontal velocity u relative to the ground.3. After release, no horizontal acceleration acts (neglecting air resistance), so horizontal velocity remains u.4. Vertically, the coin is under the influence of gravitational acceleration g downward, starting from zero vertical velocity at the moment of release.5. The horizontal position x and vertical position y as functions of time t satisfy x = u * t and y = (1 / 2) * g * t^2, which together describe a parabola when y is plotted against x.

Verification / Alternative check:
A similar situation occurs when water drops from a moving sprinkler or when a ball is dropped from a moving car. Videos or actual experiments show that the object lands ahead of the dropping point and traces a curved path relative to the ground. Textbooks on mechanics explicitly state that a body projected or dropped from a moving object becomes a projectile whose path in the ground frame is a parabola.

Why Other Options Are Wrong:
- A circular path: There is no centripetal force to bend the motion into a circle, so the path cannot be circular.- A horizontal straight line: This would require zero vertical acceleration; however, gravity acts, so the coin must fall.- A vertical straight line: This is what the boy inside the train approximately sees, because in his frame the coin has no initial horizontal velocity. The man on the ground, however, sees both horizontal and vertical motions combined.

Common Pitfalls:
Learners sometimes forget that velocities add vectorially across reference frames. From the boy perspective, the coin seems to fall straight down because both he and the coin share the same horizontal motion. From the ground perspective, the coin has horizontal motion plus vertical fall. Remember to specify the frame of reference whenever you describe motion.

Final Answer:
To the man standing on the ground, the coin appears to follow a parabolic path.