What if Earth had weaker gravity?

Throw a ball up and out. It flies in a curve, then comes back down.

Gravity pulls it down. What if gravity pulled less?

Throw a ball across a park and it flies up, curves over and drops back to the ground. The curve happens because gravity keeps pulling the ball down the whole time it is flying.

So what would happen to the same throw if Earth's gravity were weaker, say half as strong? Would the ball go a little farther, or a lot? In this lab you can change gravity in a model and throw a ball to find out.

A thrown ball's path is set by its launch speed, its launch angle and the downward acceleration of gravity, g, which is about 9.81 m/s² at Earth's surface.

This lab asks how the flight changes if g is reduced while the throw stays the same: how do the range, the highest point, the time in the air and the landing speed respond, and why?

On Earth, a good throw goes up, then comes back down.

It lands about 23 meters away.

In the model, a ball launched at 15 meters per second at an angle of 45 degrees, with no air to slow it, climbs to about 5.7 meters, stays in the air for about 2.2 seconds and lands about 23 meters away.

All the time it flies, the ball moves sideways at a steady speed while gravity keeps changing its up-and-down speed. Together they make a smooth curve called a parabola.

With v₀ = 15 m/s, θ = 45° and g = 9.81 m/s² in a vacuum, the model gives a range of 22.9 m, a peak of 5.73 m and a flight time of 2.16 s. The horizontal velocity v₀ cos θ stays constant while the vertical velocity changes by −g every second.

With the model's air resistance switched on (a football's size and mass in sea-level air), the same throw falls short, at about 18.6 m.

In our pretend lab, Olly makes gravity half as strong.

Now the ball floats up high! It stays up longer and lands much farther away.

Set gravity to 4.9, half of Earth's. Throw the same ball in the same way. Now it climbs to about 11.5 meters, stays up for about 4.3 seconds and lands about 46 meters away: twice as high, twice as long and twice as far.

One thing does not change: with no air, the ball hits the ground at the same speed it was thrown, 15 meters per second, in both worlds. Try the Moon's gravity too, and the throw goes about 139 meters.

At g = 4.9 m/s² the model gives R = 45.9 m, a peak of 11.5 m and a flight time of 4.33 s, each almost exactly double, because each is proportional to 1/g. The landing speed stays at 15 m/s, since energy is conserved and the ball returns to its launch height.

With air resistance on, halving g raises the range only from 18.6 m to 31.7 m, because the longer flight gives drag more time to act. This is a thought experiment: Earth's gravity cannot be halved.

Gravity is the pull that brings the ball down. A weaker pull brings it down more slowly.

So the ball has more time to fly forward.

A thrown ball does two things at once. It keeps moving forward at a steady speed, and gravity pulls it down faster and faster. The ball keeps flying forward until gravity brings it back to the ground.

With weaker gravity, the ball's upward speed is used up more slowly, so it climbs higher. It also takes longer to come back down. More time in the air means more time moving forward, so it lands farther away.

Horizontal and vertical motions are independent. Time aloft is T = 2v₀ sin θ / g, and range is the constant horizontal speed times T, giving R = v₀² sin 2θ / g. Both scale as 1/g, as does the peak height v_y²/2g.

The model's limits: uniform gravity, flat ground, no spin, no wind and, when switched on, quadratic drag with a constant coefficient. Real footballs and baseballs spin, and their drag changes with speed, so their real flights differ from this smooth model.

The Moon's gravity is much weaker than Earth's. A ball thrown there would fly a very long way.

Mars has weak gravity too.

We cannot change Earth's gravity, but other worlds have their own. Gravity on the Moon is about one sixth of Earth's, and on Mars a little over a third. In the model, our 15 meter-per-second throw would land about 61 meters away on Mars and about 139 meters away on the Moon, with no air.

Mars does have air, but it is very thin. In the model, switching on Mars's thin air barely shortens the throw at all.

NASA's fact sheets give surface gravity of 1.62 m/s² for the Moon and about 3.7 m/s² for Mars, 0.165 and 0.38 of Earth's. The model throw reaches 138.9 m on the Moon and 60.6 m on Mars in a vacuum.

Mars's surface air density is about 0.02 kg/m³, compared with 1.2 kg/m³ at sea level on Earth. With that density, drag in the model shortens the Mars throw only from 60.6 m to about 60.0 m.