Projectile motion

A ball thrown at 20 m/s and 45° stays in the air for 2.9 seconds and lands about 41 m away.

Two motions at once

A thrown ball, a jet of water from a hose and a long jumper all follow the same kind of curved path, a parabola. The curve looks complicated, but it is two simple motions happening together. Sideways, nothing pushes or pulls the ball once it has left your hand (ignoring air), so its horizontal velocity stays constant. Up and down, gravity accelerates it at g, exactly as in free fall. The two motions do not affect each other. Galileo described this in the fourth day of Two New Sciences, and it is still the starting point in every physics textbook.

You can test this with two coins. Flick one off the edge of a table while letting the other drop from the same edge at the same moment. You will hear them hit the floor together. The flicked coin travels sideways as well, but its downward motion is the same free fall as the dropped coin.

Splitting the launch velocity

If a ball leaves the ground at speed v and angle θ above the horizontal, its velocity splits into a horizontal part v_x = v cos θ and a vertical part v_y = v sin θ. At 20 m/s and 45°, both parts are 20 × 0.707 ≈ 14.1 m/s. From there the rest follows from free fall. The ball rises until v_y reaches zero, which takes v_y ÷ g, and the same time again to come down. So the time in the air is T = 2 v sin θ ÷ g. The highest point is H = (v sin θ)² ÷ 2g. The range is horizontal speed × time = v cos θ × T, which tidies up to R = v² sin 2θ ÷ g.

Put in the numbers for 20 m/s at 45° on Earth: T = 2 × 20 × 0.707 ÷ 9.81 ≈ 2.88 s, H ≈ 10.2 m, R = 400 × sin 90° ÷ 9.81 ≈ 40.8 m. These are the values the lab shows when it opens.

What the range formula tells you

Because sin 2θ is largest when 2θ = 90°, the longest range on flat ground comes at 45°. Angles that add up to 90° give the same range: 30° and 60° both send a 20 m/s ball about 35.3 m, but the 60° throw goes higher and stays up longer. The v² in the formula means speed matters a lot: double the launch speed and the range becomes four times longer. And g sits underneath: on the Moon, where g is about one sixth of Earth’s, the same throw goes about six times as far. Try 20 m/s at 45° on the Moon in the lab and you get about 247 m.

These formulas assume level ground, a launch from ground height and no air. Real balls lose some range to air resistance, which is the subject of the next lesson.

Sport is full of projectiles. A long jumper takes off at a lower angle than 45°, about 21° in video measurements of world-class jumpers, because running speed is easy to keep but jumping up steeply is hard, and the body lands lower than it took off. A basketball shot rises to a peak and drops into the hoop at a steep angle, which gives it a wider target. In each case the same two rules apply: steady sideways motion and free fall up and down.