If a player kicks a soccer ball in a high arc toward the opponent's goal will the velocity and acceleration of the ball be zero at the highest point in its trajectory?

Answer 1

Neither the acceleration nor the velocity of the ball will be zero during its flight.

The motion of a kicked soccer ball is actually rather complex, and we'll have to simplify the system a bit and make some assumptions if we're going to be able to talk about it here without using advanced mathematics or complex computer simulations.

So let's get those simplifications and assumptions out of the way, shall we?

We will ignore the effects of the air on the ball. In other words, we're assuming the ball is traveling through a vacuum. This is important for two reasons. One is we can assume the ball's horizontal velocity doesn't change from the moment it leaves the player's foot to the moment it lands in the back of the net or the goal tender's hands. This is because we're ignoring the effects of air resistance.

The second is we can forget about the cool "bending" tricks that great players (like Beckham) can make the ball do by putting spin on it. So, for our discussion, the ball moves in two dimensions: the vertical (the y direction) and the horizontal (the x direction). We can ignore the third direction: the z direction. Hence, the ball follows a perfect parabolic arc in one plane.

When the ball leaves the player's foot, it has an initial velocity that has both vertical and horizontal components. The horizontal component propels the ball down the field; the vertical component propels the ball up into the air. Let's look at the vertical component, first.

From the split second the ball leaves the player's foot, the ball experiences an acceleration in the downward direction. Yes, downward! Even though the ball is heading up, the acceleration it experiences is down! That is because the acceleration due to gravity is always pointed toward the center of the Earth -- in other words, DOWN! This causes the ball to slow down, but only in the vertical direction. The vertical velocity decreases until the ball reaches its apex, at which point the vertical velocity is zero. After it reaches its apex, the ball starts to accelerate downward. In other words, it picks up speed in the downward direction.

How about the horizontal component? Well, since we're ignoring air resistance, it doesn't change for the whole flight. No kidding. If the ball's horizontal velocity is 30 meters per second as it leaves the player's foot, it will be 30 meters per second for the whole flight. So, you can see that the vector sum of the vertical and horizontal components of the velocity is never zero, since there is always a non-zero horizontal component.

The discussion of acceleration is trivial. The acceleration of gravity is always there. It never goes away -- let's HOPE it never goes away!! -- so the ball always experiences it. And it's always directed downward, even when the ball's going up.

To summarize, the acceleration and velocity of the ball is always non-zero.

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1. This is somewhat counter-intuitive and hard for new physics students to get their brains around. The acceleration of gravity is perpendicular (orthogonal) to the horizontal motion. Therefore, there is no horizontal component of the acceleration due to gravity. Because there is no horizontal component, it cannot affect the horizontal motion of the ball.

2. The spin on a ball in a medium like air creates pressure differentials that produce forces on the ball that make it do funny things. The same principle is at work when a Baseball pitcher throws a curve ball, slider, or split-fingered fastball: the spin makes the ball move in the direction of rotation.

3. The ball's vertical velocity is zero only for an instant, an instant so brief as to be too small to quantify. The velocity at any instant that is too small to quantify is called the instantaneous velocity. (That should be easy to remember.)

4. By now you may wondering why I keep saying "as it leaves the player's foot" or "a split second after the ball leaves the player's foot." Well, that is because we're only talking about the motion of the ball after the player has finished doing work on the ball -- that is, KICKING it! It's a whole other physics problem to analyze what happens to the ball during the act of kicking it. The force applied to the ball during the kick is what accelerates it and gives it its initial velocity for the problem being discussed here. That acceleration has NOTHING to do with the acceleration of gravity.