I’ve introduced the idea of an *inertial frame of reference*. This is when we, and objects in our frame, are either standing still or moving with constant (straight-line) motion. In this situation, we can’t tell if we’re really moving or standing still relative to some other frame of reference. In fact, the question is meaningless.

I’ve also introduced the idea that objects moving within our frame — moving (or standing still) along with us, but *also* moving from our perspective — move *differently* from the perspective of other frames. Specifically, the speed appears different.

Now I’ll dig deeper into that and introduce a crucial exception.

Let’s start with some terminology.

*Speed* is a (single) numerical quantity that always combines two ideas: distance and time. A given speed is the rate of *some amount of distance* per *some amount of time*. The “per” means division, so speed is simply distance ÷ time.

To make things easier, the time is usually expressed as a single unit: per (one) hour, or per (one) second, or per (one) milli-jiffy. The amount of distance (miles, meters, cubits) in that time is the speed. For example, 60 MPH means 60 miles distance in one hour of time (likewise: 96 KPH means 96 kilometers in one hour).

*Velocity* is speed *in a specified direction*. As such a velocity properly includes extra numbers defining the direction. (Math fans will tell you that speed is a *scalar*, velocity is a *vector*.) Here we’ll be more general with direction. *Forward*, *backward*, and so forth, is sufficient for our purposes.

We return to Al and Em. As before, Al is on the ground next to a train track, and Em is on the train that passes by (from left to right).

As Em goes past she throws two baseballs, one forward — *with* train’s motion — and one backward — *against* the train’s motion. We’re going to consider the difference between what each of them see.

In our first example, Em passes Al at **80 MPH** (metric readers can imagine * KPH* throughout). She lobs the baseballs at a fairly slow

**30 MPH**. She throws the cyan-colored one forward and the red-colored one backward.

From Em’s perspective, she sees the both balls moving away from her at **30 MPH**. But Al sees Em’s whole frame of reference moving at **80 MPH**, so this velocity is part of what Al’s picture!

The forward-moving baseball (cyan) has a velocity in the same direction as the train, so the speeds add together (**80 + 30**). Al sees that ball moving at **110 MPH**.

The backward-moving baseball (red) has a velocity in the *opposite* direction from the train, so its speed is subtracted from the train’s (**80 – 30**). Al sees that ball moving at **50 MPH**.^{[1]}

Since the speed of both balls is less than the train’s, Al sees both balls moving “forward” (left-to-right, along with the train). Let’s reverse that situation:

Now the train is moving much slower than the balls. (Em has a pretty good arm throwing a fastball at **80 MPH** — like Mo’ne Davis!^{[2]})

The equations are the same, the forward-moving ball’s speed adds to the train’s, and **30 + 80** is the same as **80 + 30**, so Al still sees the cyan-colored ball zipping along at **110 MPH**.

Likewise, the backward-moving ball’s speed subtracts from the train’s (**30 – 80**)^{[3]}, and this gives us a negative answer: **-50 MPH**. The minus sign just means Al sees that ball as moving *backwards* at 50.

Of course, Em sees both balls moving at 80 MPH just as she threw them. Now suppose the train and the balls have the *same* speeds:

Both the train and the balls are moving at **60 MPH**. As always, Em sees the balls moving away from her at the speed she threw them.

Al still adds the train’s speed with the forward-moving ball’s speed (**60 + 60**), so he sees the cyan-colored ball moving at **120 MPH**.

He also still subtracts the backward-moving ball’s speed from the train’s (**60 – 60**^{[4]}), and this time he gets zero — **0 MPH**!

Which means, from Al’s perspective, that ball has no speed. If it were free from the train’s influence — if it left Em’s frame of reference and entered Al’s — it would drop straight to the ground.

Mythbusters performed this experiment, and it’s worth watching the clip. Just basic physics, but it’s pretty cool to see:

We’ve considered baseballs thrown with velocities that either directly add to, or subtract from, the train’s. You might wonder what happens if Em throws a ball in some other direction:

Last time we considered balls dropped straight down with gravity providing the speed. Al’s calculation of the ball’s speed then required Pythagoras and his famous theorem.

Em’s thrown baseball here is the same thing. In fact, our additions and subtractions above are reductions of the Pythagorean formula made possible by collapsing one side of the triangle to zero.^{[5]}

Okay, are you ready for the kicker?

This business of adding velocities applies to everything with one very important exception: light. *Light always moves at the same speed regardless of your frame of reference!* We call that speed: ** c**.

The situation described in *diagram 5* is different from the first four. In this case Em is shining a *light* forward and backward as she passes Al (although, remember, she thinks Al is passing her). Em isn’t throwing baseballs now — she’s throwing photons (cyan- and red-colored ones).

We’ve given the train a high speed to make the point that it doesn’t matter what the speed is. Al sees the photons moving the same speed Em does: at ** c**.

No matter how fast she zips by, *both* always see light moving at ** c**.

This is the crucial difference between *Galilean Relativity* (where velocities are added and which the ancients understood) and Special Relativity — which is Einstein’s gift.

As we’ll see, accepting this fact has some unusual circumstances.

But to fully understand them, I need introduce you to what are called *time-space diagrams*. They’re not that hard to understand, but they are abstract enough to require some explanation.

That will be our primary topic next week. Next time I’ll review what I’ve covered so far, touch a bit more on ** c**, and set the context for Special Relativity.

^{[1]} Technically speaking, the ball’s velocity is **-30** because we’re considering the train’s velocity to be **+80** (this is part of what we mean by velocity having a direction).

In reality, we’re adding in both cases, so the two equations are really **(+80)+(+30)** and **(+80)+(-30)** but I wanted to keep things simple.

^{[2]} Slightly better, actually, since Mo’ne tops out around 70 MPH — still about 10 MPH faster than any of her peers (the boys included). There is some chance she’ll be the first woman to play in the MLB. Wouldn’t that be something!

^{[3]} Technically: **(+30)+(-80)**. (*See [1] above.*)

^{[4]} See above. (*What can I say. I love footnotes!*)

^{[5]} If: **a ^{2}+b^{2}=c^{2}**, and

**b=0**, then

**b**, so

^{2}=0**a**, which is

^{2}+0=c^{2}**a**, and that means

^{2}=c^{2}**a=c**. (

*Whew!*)

March 19th, 2015 at 7:57 pm

Interesting…!

March 20th, 2015 at 10:46 am

Because…?

March 20th, 2015 at 6:51 pm

I have never came up with such a detailed analysis using known science theories and explained such analysis in a layman’s term the way you did. It’s quite an information load… But it is good that someone like you took time to explain it in such fashion.

March 20th, 2015 at 7:05 pm

And we’re just getting started here! Thanks; I’m glad you enjoyed it!

March 20th, 2015 at 9:33 am

Nice job, once again, and cool video. I didn’t quite believe that the ball dropped straight down until I saw the slow motion clip. That was pretty neat.

March 20th, 2015 at 10:51 am

Thanks. I know what you mean. In real-time it doesn’t look like it fell straight down, but the slo-mo blows you away!

Did you watch the bowling ball and feather drop in the vacuum chamber that I linked to Wednesday? That was pretty cool, too!

March 20th, 2015 at 1:11 pm

No, I’ll have to check that out!

March 20th, 2015 at 1:18 pm

You’re right…that was awesome!

March 20th, 2015 at 1:43 pm

Yeah it is! I’ve also seen footage from when Apollo astronauts were on the moon and one of them dropped a feather and a hammer. Same thing — both fall straight down at the same speed. A slower speed, since the Moon’s gravity isn’t as strong, but the same speed. It kinda twists your mind!

March 22nd, 2015 at 2:17 pm

Of course, I did like Speed movies and also, can see why baseball uses velocity, but I tried to retain some of the science in this post but may have to come back another day to make a ‘smarter’ comment! Smiles!

March 22nd, 2015 at 2:34 pm

Well, it might make more sense of you start at the beginning with Monday’s post. The whole week builds to the conclusions of Thursday and Friday.