What if you threw a ball at a lower angle?

When you throw a ball, you can aim it high or low.

Olly usually throws at a slant. What if he throws it flatter, closer to the ground? Will it go farther?

Every throw has two parts you choose: how fast the ball leaves your hand, and the angle you aim it at. Aim straight up and the ball comes back down on your head. Aim flat along the ground and it hits the ground almost at once. Somewhere in between is the throw that goes farthest.

In this lab the ball always leaves the launcher at the same speed. Only the angle changes. Start at 45°, then try a much flatter 15°, and watch how far the ball lands.

A projectile's launch velocity splits into horizontal and vertical components, v₀ cos θ and v₀ sin θ. A steep throw has a large vertical part, so it stays up a long time, but little horizontal speed; a flat throw is fast sideways but lands quickly.

This lab holds v₀ = 15 m/s and g = 9.81 m/s² fixed in the projectile model, with launch and landing at the same level and air resistance off, and lowers the launch angle from 45° to 15°. The question is how range, peak height and flight time respond, and what changes when drag is switched on.

Olly throws the ball at a slant, halfway between flat and straight up.

It flies up in a big curve and lands far away.

In the model, the ball leaves the launcher at 15 meters per second, aimed at 45°, halfway between flat and straight up. It climbs to almost 6 meters, then curves back down and lands about 23 meters away, after just over 2 seconds in the air.

The curved path is called a parabola. The ball moves sideways at a steady speed the whole time, while gravity first slows its rise and then speeds up its fall.

At 45° the components are equal: vₓ = v_y = 15 cos 45° ≈ 10.6 m/s. The model gives a peak of 5.73 m, a flight time of 2.16 s and a range of 22.9 m, matching R = v₀²/g for sin 90° = 1. The ball lands at its launch speed, 15.0 m/s, because without air nothing slows its horizontal motion and its vertical speed on landing mirrors its launch.

OpenStax notes that the range formula holds only for launch and impact on a horizontal surface; the model's launch height control can break that condition, so this lab keeps it at zero.

Now Olly throws the ball flatter, close to the ground.

It zooms along but lands much sooner. It only goes about half as far!

Now aim lower, at 15°. The ball still leaves at 15 meters per second, but its path is low and flat. It rises less than a meter and lands after less than a second, about 11 meters away: only half as far as the 45° throw.

Here is a surprise: try 75°, a steep throw. It lands at almost exactly the same place as the 15° throw, about 11 meters away, but it climbs nearly 11 meters high on the way. Two different angles, one landing spot.

At 15° the model gives a peak of 0.77 m, a flight time of 0.79 s and a range of 11.5 m, half the 45° value because sin 30° = 0.5. The horizontal speed is higher (14.5 m/s), but the time aloft is much shorter.

At 75° the range is again 11.5 m (sin 150° = 0.5), with a peak of 10.7 m and 2.95 s aloft. OpenStax notes that launch angles summing to 90° give equal ranges, the lower one with the lower apex. Likewise 30° and 60° both give 19.9 m.

A flat throw goes fast sideways, but it lands very soon.

A high throw stays up longer, but goes slowly sideways. In the middle is best!

A thrown ball does two things at once. It moves sideways at a steady speed, and gravity pulls it down. How far it goes depends on both: how fast it moves sideways, and how long it stays in the air.

A flat throw has lots of sideways speed, but it hits the ground almost straight away. A steep throw stays up a long time but hardly moves sideways. At 45°, the two are best balanced, so the ball lands farthest. Our model leaves out air, spin and wind, and stops the ball where it first lands.

The flight time from level ground is T = 2v₀ sin θ / g, and the horizontal distance is vₓT = (v₀ cos θ)(2v₀ sin θ / g) = v₀² sin 2θ / g. The product sin θ cos θ peaks at 45°, and sin 2θ is symmetric about 45°, which explains the equal ranges of complementary angles.

Model limits: uniform gravity, flat ground, no spin (no Magnus curving), quadratic drag with a constant drag coefficient when air is on, and no bounce or roll after landing. Real thrown balls spin, and their drag coefficients change with speed.

Real air pushes back on a flying ball. So real balls do not go quite as far.

In air, the best angle is a little lower than halfway.

Real balls fly through air, which pushes back on them. NASA's guide to flight explains that a baseball or a soccer ball in flight feels a moderate amount of drag, so its path is not a perfect parabola. How much drag depends on the weather and the height above sea level, because both change how thick the air is.

OpenStax adds that with air resistance, the angle for the farthest throw is somewhat smaller than 45°. Switch air on in the model to see it: for a football at 15 meters per second, 45° now gives about 18.6 meters, and 43° gives a few centimeters more.

NASA Glenn's ballistic-flight page treats the drag-free parabola as a first approximation: real balls generate moderate drag that changes throughout the flight, since it depends on the square of the velocity and on air density, which varies with weather and altitude. OpenStax notes that with air resistance the angle of maximum range is somewhat smaller than 45°.

The model agrees: with drag on (a 0.43 kg football, C_d 0.25, 0.038 m²), ranges at 15 m/s are 18.52 m at 40°, 18.63 m at 43° and 18.61 m at 45°, and 15° falls to 10.4 m. In the model, drag also makes the way down steeper than the way up.