Motion in Two dimensions

People who play sports such as tennis and squash know that, no matter how hard they hit the ball, they cannot make it follow a horizontal path. The motion of a ball through the air is always affected by the Earth’s pull.

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There is a simple relationship between the total distances travelled after t, 2t etc. by an object falling vertically. Can you spot it?

The diagram shows the results of photographing the motion of a ball projected horizontally alongside one released so that it falls freely. The photographs are taken at equal time intervals.

The free-falling ball travels increasing distances in successive time intervals. This is because the vertical motion is accelerated motion, so the average speed of the ball over each successive time interval increases.

The ball projected horizontally travels equal distances horizontally in successive time intervals, showing that its horizontal motion is at constant speed. Vertically, its motion matches that of the free-falling ball, showing that its vertical motion is not affected by its horizontal motion.

When an object has both horizontal and vertical motion, these are independent of each other.

The motion of any object can be resolved in two directions at right angles to each other. The two motions can then be treated separately.

This important result means that the horizontal and vertical motions can be analysed separately:

• for the horizontal motion at constant speed, the equation v = s ÷ t applies
• for the vertical, accelerated, motion, the equations of motion with uniform acceleration apply.