We need one more thing—how about Newton’s second law? This says the acceleration depends on the net force (F_net) and the mass (m) of an object. It’s usually written as F_net = m × a, but we can rearrange it like this: a = F_net/m. Combining this with our gravitational force, we get something pretty interesting:

Courtesy of Rhett Allain

Since both gravity and acceleration depend on the mass of the ball, the mass cancels. We find that any object on Earth has a downward acceleration of 9.8 meters per second per second (m/s²). This means that if you drop a bowling ball and a marble at the same time, they’ll hit the ground at the same time—even though the gravitational force on the bowling ball is thousands of times higher. Weird, right?

Anyway, now, in the presence of gravity, if you kicked a ball at an upward angle, it’s vertical velocity would slow, halt, and reverse, with the speed increasing as it falls. In other words, it starts accelerating in the downward direction as soon as it’s kicked, even while it’s moving upward.

What about the horizontal motion? Ah, since there’s no horizontal force after the initial kick, the ball continues traveling forward at the same speed, just like in space. People tend to think a ball falls because its forward motion slows, but actually it’s the opposite. Without air drag it doesn’t slow down at all. It only stops because the ground gets in the way.

So what we get for a trajectory is that familiar upside-down parabola, often called a ballistic trajectory because it’s the path of any unpowered projectile, like a cannon ball, a bullet, or a basketball. Any flying object for which gravity is the only (significant) force acting on it will move this way.

Soccer With Air

Happily, the Earth does have air. But it drastically changes the game. Now there is a continuous force acting horizontally, which we call air resistance, or drag, and it pushes in the direction opposite to the ball’s motion.

Think of air molecules as a bunch of tiny ping-pong balls. As a soccer ball moves through the air it collides with gazillions of these little air balls, and each collision exerts a backward-pushing force; all combined, this creates the total air-resistance force. The bigger the object, the more collisions it has to fight through.