A boy sitting in a train moving with uniform horizontal velocity drops a coin out of the window. As seen by a man standing on the ground, what trajectory does the coin follow as it falls to the Earth?

Introduction / Context:
This question tests your understanding of projectile motion and relative observation from different frames of reference. When an object is dropped from a moving vehicle, it has both horizontal and vertical motion components. A passenger inside the vehicle and an observer on the ground may describe the path differently. Recognising the shape of the trajectory as seen from the ground is a standard kinematics concept.

Given Data / Assumptions:

• The train moves horizontally with uniform velocity, meaning constant speed and direction.
• A boy in the train drops a coin from some height.
• Air resistance is neglected for simplicity.
• We consider the path of the coin as seen by a person standing on the ground.

Concept / Approach:
At the instant the coin is released, it already has the same horizontal velocity as the train, because it was moving with the train. When the coin is dropped, there is no horizontal force acting on it (ignoring air drag), so it continues to move horizontally with the same uniform speed. At the same time, gravity acts downward, giving the coin a vertical acceleration. The combination of uniform horizontal motion and uniformly accelerated vertical motion produces a parabolic path, which is the characteristic trajectory of a projectile in a uniform gravitational field.

Step-by-Step Solution:
Step 1: At the moment of release, the coin has a horizontal velocity equal to the uniform velocity of the train. Step 2: Immediately after release, the horizontal component of velocity remains constant because no horizontal force acts on the coin. Step 3: Gravity provides a constant downward acceleration g, giving the coin a vertical velocity component that increases with time. Step 4: The horizontal displacement increases linearly with time, while the vertical displacement increases with time squared due to constant acceleration. Step 5: When you plot vertical displacement against horizontal displacement, the resulting curve is a parabola, which is the path seen by the ground observer. Step 6: Therefore, the man standing outside the train will find the trajectory of the coin to be parabolic.

Verification / Alternative check:
In mathematical terms, horizontal motion can be described by x = u * t, where u is constant horizontal speed. Vertical motion follows y = (1/2) * g * t^2 downward (if initial vertical velocity is zero). Eliminating time t between these two equations leads to y being proportional to x^2, which is the equation of a parabola. This confirms that the path is not a straight line but a curved parabolic trajectory. Numerous experiments, such as dropping a ball from a moving cart, show similar behaviour.

Why Other Options Are Wrong:
A horizontal straight line would require no vertical acceleration, which is impossible when gravity acts. A vertical straight line would be seen by the boy in the train if he ignores the horizontal motion, but not by the ground observer, who sees both components. A circle does not arise from combining uniform horizontal motion with uniform vertical acceleration; the equations of motion do not match circular motion conditions.

Common Pitfalls:
Students sometimes think that dropping an object from a moving vehicle makes it fall straight down because they imagine only the vertical motion. This error usually comes from forgetting that the object already has horizontal velocity at the moment of release. Another confusion arises from mixing the viewpoints of the train passenger and the ground observer. Always specify the reference frame: to a person in the train, the coin may appear to fall nearly straight down; to the ground observer, the path is clearly parabolic.

Final Answer:
To a man standing on the ground, the coin follows a parabola as it falls.