Modeling the Motion of a Tossed Ball in an Accelerating Elevator

Photo: Rhett Allain. Two views of a ball tossed in an accelerating elevator.

There is this great elevator in the Hyatt Regency hotel in New Orleans. First, it’s a fairly high acceleration elevator (some of them are just super weak). Second, it’s got a nice glass window on the side. This means you can see outside as you move up floor. But it also means that people can see outside INTO the elevator.

So, here is the fun physics question:

“What happens when you toss a ball up inside an accelerating elevator?”

I actually have a video of this situation. Check it out.

If you like, you can use video analysis to find the acceleration of the elevator and of the ball — but I will let you do that as a homework exercise.

Instead, I’m going to model this motion (with python) from two different reference frames. First, I will look at the motion of both the elevator and the ball from a stationary viewpoint outside of the elevator (this would be an inertial reference frame). Second, I will model the motion from the non-inertial reference frame inside the elevator.

Stationary Reference Frame

We have this model for forces and motion. You can call it Newton’s Second Law, or you could call it the Momentum Principle. Either way, this model says that the net force on an object CHANGES its motion. In terms of mass and acceleration, it looks like this.

In this expression, the net force is a sum of all the real forces on the object. Real forces could be something like:

  • The gravitational force between the object and the Earth.
  • A frictional force due to a surface interaction.
  • Constraint forces like the contact force with something like a cable attached to the object.

My label of “real” forces will make more sense later. But they are due to some type of interaction with another object.

The other important thing about this force-motion relationship is that the acceleration must be measured in an non-accelerating reference frame (we call this an inertial reference frame).

Technically, if you are standing on the ground and stationary (very likely) it’s NOT an inertial reference frame. Because the Earth is rotating, you are moving in a circle that takes one day to complete. That means your surroundings are also accelerating (centripetal acceleration). However, the acceleration is small enough that we can pretend it’s an inertial frame. Let’s do that.

OK, so I have two things to model in this inertial frame. There is the upward accelerating elevator and then the tossed ball with a downward acceleration of -9.8 m/s².

Since both of these objects (ball and elevator) have constant accelerations, I can just use the following kinematic equation to determine the position at each instant of time.

Yes, that’s a vector kinematic equation — you can do that. All I need is the vector value for the position (r) at time t = 0 and the velocity (v) at t = 0. Boom. That’s it. For the ball, it will have an acceleration equal to the vector g after it starts with some initial velocity.

Here’s what this would look like. I made this model in Glowscript-Vpython for a nice 3D model. You can see the code here.

Of course the 3D model isn’t always so useful. How about a graph showing the vertical position of both the ball and the bottom of the elevator. Here’s what that looks like.

Just to be clear, the red curve is for the ball and the blue curve is for the elevator. Now for two questions:

  • At what time (approximately) is the ball at its highest point? What is this position value?
  • At what time (approximately) is the ball at its highest point above the elevator? What is the maximum height above the elevator’s floor?

The answer to the first question can be found from the graph. Just look at the peak of the red curve. That’s right near the end of the curve at a time of about 1.76 seconds with a height of 0.127 meters (I started the thing with a negative y value). So, this is the highest point you would see the ball get to (before being caught) as you observe it from an inertial reference frame.

What about the height of the ball above the elevator floor? This is just the position of the ball minus the position of the floor (it’s the vertical distance between the red and blue curve above). Here is that plot.

Yes, that looks like a parabola — because it is. But this isn’t the same as the red parabola above. That one is for the motion of a ball with a vertical acceleration of -9.8 m/s². This one is different. But you can get what we need — the maximum height of around 2.0 meters at a time of about 1.4 seconds. Go look at the other graph at t = 1.4 seconds. Can you see the highest point?

Accelerating Reference Frame

Now let’s say you are looking at this ball while INSIDE the accelerating elevator. The tossed ball doesn’t seem to act like it normally does. In fact, you don’t even feel normal yourself. You feel heavier. What’s up with that?

This is a non-inertial reference frame since it’s accelerating. But that means that the force-motion model doesn’t really work (Newton’s Second Law). Just consider this — if you toss the ball, there is only one force acting on it while it’s in the air. That’s the gravitational force (m*g). This would make the ball have an acceleration of g. But that doesn’t happen.

There is a way to make a Newton’s Second Law work in a non-inertial reference frame — add a fake force. It’s a fake force because it’s not due to an interaction with another object. It’s just there to make Newton’s 2nd Law work.

In general, this fake force will have the following form.

OK, here is a quick comparison of forces in the two frames (for just the ball).

In the inertial frame, the ball has an acceleration of g (the vector) but the floor is also accelerating. Solving for the acceleration in the y-direction gives:

In the non-inertial reference frame, the floor is stationary (or at least moving at a constant velocity) but there is an extra force on the ball. Finding the acceleration again:

This acceleration in the non-inertial frame is just a constant value. I can use this acceleration along with the kinematic equation to find the maximum height of the ball tossed with an initial velocity of v0 (the velocity at the maximum height is 0 m/s).

Using an initial velocity of 5 m/s (with respect to the elevator) and an elevator acceleration of 2.5 m/s², I get a max height (from the toss point) of 1.02 meters. Looking at the graph showing the ball position with respect to the elevator, it starts at 1 meter and gets up to 2.02 meters before going back down. That’s a change in height of 1.02 meters.

Just for a comparison, here is the trajectory of the ball as seen in the elevator and from outside the elevator (inertial vs. non-inertial) so that you can see the difference in apparent acceleration.

So, it’s clear that the ball as seen from inside the accelerating elevator is not just a normal tossed ball. But you can still use this accelerating reference frame to get value you want (the max height inside the elevator).

Downward Accelerating Elevator

OK, but what about this? Suppose the elevator is accelerating DOWN with an acceleration of -9.8 m/s². What would happen then? Well, the fake force would be the exact opposite of the gravitational force.

That means the ball would have zero apparent forces and a zero apparent acceleration. It’s as if the ball had zero weight. Yes, this ball would be “weightless”.

So, if you want to think about astronauts in the International Space Station then this same idea works for them. The reference frame of the ISS is moving in a circle and accelerating (centripetal acceleration). The value of this acceleration is equal to the value of the gravitational field (g) at that altitude so that the fake force exactly cancels the gravitational force.

In their reference frame, they are weightless. Of course they aren’t weightless. There’s still the gravitational force acting on them and this force causes their circular motion around the Earth.

You could even take this one step farther. Einstein’s Equivalence Principle says that there is in fact zero difference between a gravitational field and an accelerating reference frame. If you want you could even use this to say that the gravitational force is really just a result of an acceleration. But let’s not get into that right now.

One last thing about fake forces. Yes, they can make a problem like this ball in the elevator thing easier. However, you have to be careful and always distinguish real and fake forces. I have seen many students use fake forces in the wrong way. It’s hard enough using real forces — that’s why pretty much every introductory physics textbook leaves these out. It’s probably a good idea.

Physics faculty, science blogger of all things geek. Technical Consultant for CBS MacGyver and MythBusters. Former WIRED blogger.

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