Place two coins at the edge of a table. Flick one so that it flies off horizontally. Let the other fall straight down at the same instant. Listen. There is one sound, not two. The coin that travelled a metre sideways and the coin that fell straight down reach the floor together.
The horizontal motion does not know about the vertical one
Both coins leave the table at the same height and at the same instant. Both have zero vertical velocity at the start. Both have the same downward acceleration \(g\). So both take the same time to fall through the same height. The horizontal velocity of the flicked coin has no effect on this time. The horizontal motion carries the coin sideways while the vertical motion carries it down, and neither motion changes the other.
A ball is dropped from a height. At the same instant, a second ball is thrown horizontally from the same height. Air resistance is neglected. Which ball reaches the ground first?
Watch the dashed line joining the two balls in Fig 3.1. The line stays horizontal for the whole fall. At every instant the two balls are at the same height. The thrown ball moves to the right at a steady rate while it falls, because nothing pushes it horizontally after it leaves the hand.
When a body moves in a plane, its motion along one axis does not affect its motion along a perpendicular axis. The two motions can be treated separately, each with its own initial velocity and its own acceleration. The two are linked only by the time, which is the same for both.
Think of the body's shadow on the floor and its shadow on a wall. The floor shadow shows only the horizontal motion. The wall shadow shows only the vertical motion. Each shadow moves as if the other did not exist. The real motion is the two shadows put together at the same instant.
How to use the principle
Choose two perpendicular axes, usually horizontal \(x\) and vertical \(y\). Split the initial velocity into components along the axes. Write down the acceleration along each axis. Then solve two one-dimensional problems. The only quantity that appears in both is the time \(t\). Whatever the \(x\) equations say about \(t\) is true in the \(y\) equations, and the other way round.
A coin leaves the edge of a table \(0.8\,\text{m}\) high with a horizontal velocity of \(2\,\text{m/s}\). Take \(g = 10\,\text{m/s}^2\). How long is the coin in the air, and how far from the table does it land?
Axes: \(x\) horizontal along the flick, \(y\) vertically upward from the floor. Horizontal: \(u_x = 2\,\text{m/s}\), \(a_x = 0\). Vertical: \(u_y = 0\), \(a_y = -10\,\text{m/s}^2\), starting height \(0.8\,\text{m}\).
Vertical: \(y = 0.8 + 0 \cdot t - 5t^2\). The coin lands when \(y = 0\). Horizontal: \(x = 2t\).
\(0 = 0.8 - 5t^2\), so \(t^2 = 0.16\) and \(t = 0.4\,\text{s}\). Then \(x = 2 \times 0.4 = 0.8\,\text{m}\).
The coin is in the air for \(0.4\,\text{s}\) and lands \(0.8\,\text{m}\) from the table. A coin dropped from the same edge would also take \(0.4\,\text{s}\); it would simply land at \(x = 0\).
Two balls are thrown horizontally from the same height, one at \(5\,\text{m/s}\) and the other at \(10\,\text{m/s}\). Air resistance is neglected. Which statement is correct?
A rifle is fired horizontally, and at the same instant a bullet is dropped from the same height beside the rifle. Air resistance is neglected. Which bullet hits the ground first?
Before Galileo, a cannon ball was drawn as travelling in a straight line, then dropping almost vertically. Galileo rolled balls off the edge of a table and marked where they landed for different speeds. He reasoned that the horizontal motion continued unchanged while the vertical motion was the same accelerated fall he had measured on his inclined planes. Putting the two together, he showed that the path must be a parabola. It was the first time two independent motions had been combined to predict a curve, and it is the method you use in the next lesson.
A coin is flicked off a desk \(0.75\,\text{m}\) high and lands \(1.5\,\text{m}\) away. Roughly what was its horizontal speed in m/s? Take \(g = 10\,\text{m/s}^2\).
- Believing that a faster horizontal throw stays in the air longer.The time in the air comes from the vertical motion only. Horizontal speed decides where, not when.
- Using the horizontal speed as the initial vertical velocity.Split the initial velocity into components first. A horizontal throw has \(u_y = 0\).