1.3 Falling Objects

Critical Questions:

  • How does something move when it’s falling through the air?
  • What exactly happens when you throw a ball up and it falls back down?

Before we leave this chapter behind and start getting into the real meat of physics, I’d like to discuss one more topic: falling objects. This will complete our picture of simple motion and set the stage for the chapter on Newton’s Laws.

Imagine that you hold a ball in your hand. Picture one that you feel familiar with — a baseball, or a tennis ball, or your dog’s little red plastic chew ball (better yet, pick up an actual object and use that instead). Now imagine holding that ball out in front of you and letting go.

Dog fetching ball
Warning: the presence of an actual dog may affect the outcome of this experiment.

If you’ve read along carefully so far, you should feel quite confident in describing the ball’s motion: it starts with no velocity, then accelerates downwards. Easy.

You’ve just borne witness to gravity, a force so common to everyday experience that we usually don’t even notice it’s there until the front end of our car is dangling precariously over the edge of the cliff and we have to edge slowly backwards so that it doesn’t tip. The truth is, however, gravity is a fantastic mystery to physicists. It is the least understood of the four fundamental forces; it seems to play by its own rules and tends to go off and do things like create black holes or twist the fabric of space-time.

But we were talking about motion, and so once again I’m going to ask you to hold onto your questions about gravity — maybe write them down on a scrap of paper and trust me when I say that they will be answered. For now, let’s just say that the falling ball is undergoing constant downward acceleration as it drops from your hand to the floor.

Here’s a tricky question: what happens if you throw the ball up in the air and then catch it? If your answer was, “it goes up and then it comes down,” you are not yet thinking like a physicist.

First of all, let’s be very specific about the time we’re talking about. We will consider only the time between the moment the ball leaves your hand and the moment it lands on your hand again. In other words, we’re going to ignore whatever your hand does to the ball, and consider a ball which is already moving upwards through the air.

We can now begin to describe the ball’s motion more accurately. We know that it is initially moving upwards, so we can say it has some initial velocity. It then slows down more and more, until it eventually starts to fall down again. It so happens that this second part, when the ball moves downwards, looks just like what happened when we simply dropped the ball from our hands, and we know that that is a case of downward acceleration. But what about when the ball is moving upwards? And what happens in that moment between its upwards and downwards motion — does the ball stop moving? Stop accelerating?

To answer these questions, let’s imagine that the ball has its own little speedometer. As we said, the ball is initially moving upwards, so the speedometer starts out pointing at some number — say, 40 km/h[1]. As the ball moves upwards, it slows down — the speedometer needle starts to fall. As I just said, this is a case of constant acceleration, so the needle moves at a constant rate. Picture that needle as it falls past 30 km/h, then 20, then 10…

What happens next? In a car, the needle would have to stop at zero, because there’s nothing slower than zero.[2]

But our tossed ball doesn’t come to a nice, easy stop at the top of its trajectory. Because it’s under the influence of gravity, it’s must be constantly accelerating (this sentence might be more believable to you once you’ve read the next chapter on forces). The ball’s speedometer needle moves steadily down to zero and doesn’t stop. So what’s beyond zero? Well, negative numbers. And in physics, negative numbers imply an opposite direction.

Lo and behold, that sounds like exactly what the ball ends up doing: it changes direction.

What happened in the meantime is that the ball’s speed momentarily passed through the zero mark. The ball stopped moving, but only for a moment, and in the meantime it never stopped accelerating (the needle kept moving).

Speedometer with negative numbers
The needle points at the speed; the speed of the needle is the ball's acceleration.

Saying that the ball stopped momentarily but did not stop accelerating is just the same as saying that at some point it was moving at 20 km/h, even though the needle passed through that point on the speedometer without stopping.

Going further, we could say that the entire time the ball was in the air, it was accelerating downwards — even when it was moving up. We know this is true because the ball’s speedometer needle never stopped falling during the entire throw.

Congratulations! You’re still reading this, despite the headache you have from the preceding paragraphs. And you’ve made it through your first difficult physics concept. If you’re sweating, that’s okay, and I highly recommend going back a bit and reading it again until you feel comfortable. If you found it easy, that’s okay too, but there’s more to come.

Big Ideas:

  • If something is in the air, it is accelerating downwards due to gravity.
  • This is true whether it’s moving up, moving down, or even stopped momentarily at the peak of its trajectory. It always has a constant downwards acceleration.
  1. Please note that this is much faster than you should throw your ball, unless you’re outdoors and well away from any car windows, small animals, or other people.
  2. In fact, if you were in your car and you saw your speedometer doing this, you would experience a very sudden stop. That’s because when we brake, we use non-constant acceleration: we gently ease up on the brake pedal, allowing the car to roll placidly to a stop. The next time you’re sitting in the passenger seat, watch the speedometer as you pull up to a red light. You’ll find that it falls quickly at first, and then falls more and more slowly as it nears zero.

2 thoughts on “1.3 Falling Objects

  1. I disagree with you when you say the ball “stopped for a moment at the top of the trajectory”. For a time zero the ball had zero velocity, yes, but you cannot describe the time zero as a “moment” implying a “time” of greater than zero. So, in my opinion, although the vertical velocity was indeed zero at one “point”, it was not at that “point” for more than a “point in time” which cannot be greater than zero time. The ball was actually at the apex “point” for the exact same amount of time that it was at any other point along the path. So, you would have to ask yourself how long did the ball remain at any other point? The truth is that the ball is in constant motion and under constant acceleration while it in not being acted upon by another force, hence, it cannot “stop”, otherwise it would be in violation of Newton’s first law of motion. At the apex, the ball changes direction (sign changes)but does not stop moving. In order to described that something has indeed stopped you would have to declare how long it had stopped. It would have to have a time stopped other than zero time, otherwise it could not be described as stopped. Since the time of zero velocity at the apex is zero, the ball had not stopped!

    1. I think this is another case where physics and everyday language evolved for very different purposes. It’s true that saying “the ball stopped for a moment” may not perfectly capture the phenomenon we want to describe, but in some ways it is correct – as long as we can use ‘moment’ to describe a point in time. Other descriptions, like “the ball’s velocity passed through zero” may work, but they sound worse. When talking about physics using words instead of math, a certain amount of imprecision will always be unavoidable!

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