What happens if we toss a coin in a moving train?

The Backward-Curving Coin: Newtonian Physics on a Moving Train

We’ve all seen the classic physics thought experiment: a ball dropped from a moving train. It falls straight down from the perspective of someone on the train, but to a stationary observer, it follows a parabolic arc. But what happens when we replace the ball with a coin, and instead of a constant velocity train, we introduce acceleration? The outcome is less intuitive and surprisingly demonstrates a fundamental principle of Newtonian mechanics.

Let’s imagine a scenario: you’re on a train accelerating from a standstill. You hold a coin in your hand, initially at rest relative to the train. You then toss the coin straight up. What appears to happen?

From your perspective inside the accelerating train, the coin will appear to curve backward as it falls. This isn’t some magical effect; it’s a straightforward consequence of inertia.

When you toss the coin upwards, it inherits the train’s initial velocity at that moment. Crucially, the coin doesn’t continue to accelerate along with the train. The only force acting on the coin (neglecting air resistance for simplicity) is gravity, pulling it downwards. The train, however, is constantly gaining speed. This difference in acceleration is the key.

While the coin travels upwards, the train is speeding up. By the time the coin reaches the apex of its trajectory and begins to fall, the train is already moving significantly faster than the coin. The train is effectively pulling away from the coin, causing it to appear to lag behind and arc backwards. To an observer on the ground, the coin would still follow a parabolic trajectory, but the apex of that parabola would be further back than might be expected if the train were at a constant velocity.

This phenomenon highlights the concept of inertia – an object in motion tends to stay in motion with the same velocity unless acted upon by a net force. The coin, once tossed, has no horizontal force acting upon it beyond the initial velocity given by the toss. The train, however, does have a horizontal force acting upon it (the engine’s power). This difference leads to the seemingly backward curving trajectory.

Therefore, the next time you find yourself on an accelerating train, consider the humble coin toss. It’s a simple experiment that powerfully demonstrates the interplay of inertia, acceleration, and relative motion – a microcosm of Newtonian physics in action. The backward curve isn’t an illusion; it’s a clear visual representation of the difference between the coin’s constant horizontal velocity and the train’s increasing horizontal velocity.