How Free Fall and Projectile Motion Work: Kinematics

Free fall: a steady gain in speed

When nothing but gravity acts on an object, we call its motion free fall. Near the Earth's surface the downward acceleration is about 9.8 m/s², written g. That means a dropped ball gains roughly 9.8 metres per second of speed every second.

So the speed after t seconds, starting from rest, is

v = g t

After one second the ball moves at about 9.8 m/s, after two seconds about 19.6 m/s, and so on. Plot speed against time and you get a straight line through the origin. The slope of that line is the acceleration, which is why a straight speed graph means the acceleration is constant.

Why distance is a different story

Speed grows in step with time, but distance does not. The ball is moving faster every moment, so each second covers more ground than the last. Adding up all those growing steps gives

d = ½ g t²

The distance depends on time squared. In the first second a ball falls about 4.9 m. In two seconds it has fallen four times as far, about 19.6 m. A graph of distance against time is a curve that steepens, and its steepness at any instant is the speed.

Does heavier fall faster?

Everyday experience says yes: a hammer beats a feather. But the feather loses because of air, not because it is light. Air pushes back on an object, and that push matters a great deal for something wide and fluffy.

Without air, every object falls with the same acceleration. Gravity pulls harder on a bigger mass, but a bigger mass is also harder to speed up, and the two effects cancel exactly. Galileo reasoned his way to this idea centuries ago, and astronauts later demonstrated it on the Moon by dropping a hammer and a feather together.

The lab uses the ideal version of this: no air resistance, constant g. That is a simplified teaching model, and it is a very good one for compact, dense objects over short drops.

Throwing straight up

Toss a ball upward and gravity does not care which way the ball is moving. It still pulls down at 9.8 m/s². So the ball slows on the way up, stops for an instant at the top, then speeds up on the way down.

Two neat facts follow from this symmetry:

• the trip up takes exactly as long as the trip back down to the launch height, and
• the ball returns to your hand moving at the same speed it left, just in the opposite direction.

The height gained depends on the launch speed v as h = v² / (2g). Double the launch speed and the ball climbs four times as high, because the speed is squared.

Projectiles: two motions at once

Now throw the ball at an angle. The path is a curve, but the secret is to stop thinking about the curve and split the motion into two independent parts.

• Horizontal: nothing pushes sideways (ignoring air), so the horizontal speed stays constant.
• Vertical: gravity acts exactly as it does in free fall, pulling down at g.

These two parts do not affect each other. A ball rolled off a table and a ball simply dropped from the same height hit the floor at the same moment, even though one travels sideways as well. The sideways speed only decides how far it lands; the fall time is set by the height alone.

Splitting the launch velocity

An angled launch at speed v and angle θ has a horizontal part v cos θ and a vertical part v sin θ. Only the vertical part fights gravity, so it controls the flight time and the peak height. At the very top, the vertical speed is zero but the horizontal speed is unchanged, which is why the ball is still moving sideways at its highest point.

Range and the 45° rule

On level ground, the horizontal distance a projectile covers is

R = v² sin(2θ) / g

The angle only enters through sin(2θ), and that function is largest when 2θ is 90°, so 45° gives the longest range. Angles that add up to 90°, such as 30° and 60°, land at the same spot: the lower one flies fast and flat, the higher one flies slow and tall.

Speed matters more than angle. Because v is squared, doubling the launch speed multiplies the range by four. Real throws bend away from this neat rule once air drag and launch height come in, but the idea of two independent motions stays the foundation.

Common misconceptions

• "The ball keeps a force from your hand." After release, only gravity (and a little air) acts. The ball keeps moving because of its velocity, not because of a stored push.
• "At the top of the arc, the ball is not accelerating." Its vertical speed is zero for an instant, but the acceleration is still 9.8 m/s² downward.
• "Heavy things fall faster." In a vacuum they do not. Air resistance is the only difference.
• "Speed and acceleration are the same thing." Speed is how fast you move; acceleration is how fast the speed is changing. A ball at the top of a throw has no vertical speed but plenty of acceleration.

How to explore this in the lab

The Forces & Motion lab runs a real physics engine, so you predict first and then watch the result. A good path through the free first world:

1. Begin with Free Fall to see speed climb at a steady rate.
2. Check the squared law in How Far It Falls.
3. Add sideways motion in Horizontal Launch, then fire at an angle in Projectile Range.
4. Test the Galileo idea with Mass Doesn't Matter.

The lab also has a free lesson on floating and sinking, Float, sink, or stay suspended?, if you want to see another force at work. Remember that the simulation is a simplified model with constant gravity and no air drag, so it shows the core behaviour cleanly rather than matching a real stadium throw.