What if a ball were thrown twice as fast?

Olly throws a ball and it lands over there.

Now he throws it twice as fast. Will it go twice as far? Or even farther?

If you ride your bike twice as fast for the same time, you go twice as far. So it seems natural to guess that a ball thrown twice as fast lands twice as far away.

But a thrown ball is not like a bike on a road. It flies up, then gravity pulls it down, and how long it stays in the air depends on the throw too. In this lab you can launch a ball at 10 meters per second and then at 20, always at the same angle, and measure how far each one goes.

OpenStax states that, on level ground with air resistance neglected, a projectile's range is directly proportional to the square of its initial speed and inversely proportional to the acceleration of gravity.

This lab tests that claim in the projectile model. The launch angle stays at 45°, launch and landing heights are equal, gravity is Earth's 9.81 m/s² and air is off; only v₀ changes, from 10 to 20 m/s. Watch three outputs: range, highest point and time in the air. Then switch air on and see whether the square law survives.

First Olly throws the ball gently, at a slant.

It flies up in a little curve and lands not very far away.

In the model, a ball launched at 10 meters per second at 45° climbs to about 2.5 meters, stays in the air for about 1.4 seconds and lands about 10 meters away. That is a gentle throw, like an easy toss across a garden.

Throughout the flight the ball moves sideways at a steady speed of about 7 meters per second, while gravity slows its rise and then speeds up its fall.

At v₀ = 10 m/s and 45°, both components are 7.07 m/s. The model gives a flight time T = 2v_y/g = 1.44 s, a peak h = v_y²/2g = 2.55 m and a range vₓT = 10.2 m, equal to v₀²/g. The ball lands at 10.0 m/s, its launch speed, as expected without drag.

Each of these quantities depends on the launch speed in its own way, and that is what makes the next step interesting: time depends on v_y alone, while range multiplies vₓ by T.

Now Olly throws the ball twice as fast.

It flies much higher and stays up longer. It lands four times as far away!

Now launch the ball at 20 meters per second, twice as fast, still at 45°. It climbs to about 10 meters, four times as high. It stays up for about 2.9 seconds, twice as long. And it lands about 41 meters away: four times as far as the gentle throw.

Try three times as fast, 30 meters per second. The ball lands about 92 meters away, nine times as far. Each time, the distance grows by the speed multiplied by itself.

At v₀ = 20 m/s the model gives R = 40.8 m, h = 10.2 m and T = 2.88 s: ×4, ×4 and ×2 against the 10 m/s throw. At 30 m/s, R = 91.7 m (×9) and T = 4.32 s (×3). At 5 m/s, R = 2.55 m (×¼).

Range and peak height both scale as v₀², flight time as v₀. The landing speed equals the launch speed each time, 20.0 m/s here, because the model has no drag to remove energy. The factor of four holds at any fixed angle, since sin 2θ is common to both throws: at 30° the ranges are 8.83 m and 35.3 m.

A fast throw does two things. It goes faster sideways. And it stays up in the air longer.

Twice as fast, and twice as long: that makes four times as far!

Throwing faster helps in two ways at once. The ball moves sideways twice as fast, so in each second it covers twice the ground. And it is also thrown upward twice as fast, so it takes twice as long for gravity to bring it back down.

Twice the sideways speed for twice the time gives four times the distance. Our model is simple: it ignores air unless you switch it on, keeps the ground flat and stops the ball where it first lands.

With the angle fixed, both components scale with v₀. The flight time T = 2v₀ sin θ / g is proportional to v₀, and the range is vₓT = (v₀ cos θ)(2v₀ sin θ / g), proportional to v₀². The peak height v_y²/2g also scales as v₀², which is why it quadruples too.

Model limits: uniform gravity and flat, level ground; no spin, so no curving; quadratic drag with a constant drag coefficient when air is on; and the ball stops where it first touches the ground. The ball is drawn larger than to scale.

Real air pushes back on a flying ball. A fast ball gets pushed back even more.

So in real air, twice as fast does not quite go four times as far.

Real balls fly through air. NASA's guide to flight explains that drag depends on the square of the speed, so a ball going twice as fast feels about four times the push from the air, and that push keeps changing during the flight.

Switch air on in the model to see what that means. For a football at 45°, the gentle throw now lands about 9.2 meters away and the throw twice as fast about 29 meters away. That is a little over three times as far, not four: the faster ball loses more to the air.

NASA Glenn describes drag as depending on the square of the velocity and on air density, which is why ballistic flight is only a first approximation for a baseball or a soccer ball. OpenStax adds that when air resistance is considered, the angle for maximum range is somewhat smaller than 45°.

In the model with drag on (0.43 kg, C_d 0.25, 0.038 m²), 10 m/s gives 9.22 m and 20 m/s gives 29.1 m at 45°, a ratio of 3.16. At 20 m/s the best angle is about 42°, at 29.19 m. Air also lowers the landing speed: 14.8 m/s for the 20 m/s throw.