Topic 1.2: Displacement, Velocity, and Acceleration

346.95

This is a Top Fuel dragster, with about eleven thousand horsepower sitting right behind the driver’s head, and that smoke is the burnout, the driver spinning the rear tires to heat them up so they’ll grip. In September 2026, Shawn Langdon made the fastest run in NHRA history, and we’ve rebuilt it here from the timing data, so watch closely, because it’s over in less than four seconds.

That’s a thousand feet from a standing start in three point six six seconds, crossing the line at almost three hundred fifty miles an hour, so on average, every second of that run added about a hundred miles an hour to the car’s speed. When it’s over, the track hands the team a little strip of paper. It’s called a timeslip, and it has just a few distances, times and speeds printed on it.

By the end of this lesson, you’ll be able to take a timeslip like this one and work out how fast that car was going, and how hard it was accelerating, stretch by stretch down the track, and you’ll know exactly what it takes to catch its speed at a single instant. And that’s a skill the AP exam tests all the time, because a timeslip is really just a table of positions and times, and AP questions love to hand you a table like that and ask for an average velocity or an average acceleration.

Before we decode a single number, I want you to make a bet about that run, because the timeslip is going to settle it. Where was the car’s velocity biggest, and where was its acceleration biggest? Before we can talk about how fast anything is going, we need to agree on where it is, and with a car this long, that’s a better question than it sounds.

Where is the car?

When we say the car is on the starting line, which part of it do we mean? The nose sticks way out front, the driver sits in the middle, and the rear tires are more than twenty feet behind the front ones. The timing system settles it for us, because its beams run low across the track, right where the front tires roll through them, so as far as the clock is concerned, the whole car is one point at its front tires.

Physicists make that move on purpose and call it the object model, ignoring the car’s size and shape and whatever is going on inside it, and treating it as one point that still has all of the car’s mass. Once the car is a point, we can give it a position, so let’s put zero right on the starting line and make the direction toward the finish positive, which puts the finish line at positive three hundred four point eight meters.

What we really care about is how much that position changes, and we call it the displacement, which is the final position minus the starting position, delta x equals x minus x naught. The triangle is the Greek letter delta, which means the change in something, and the little zero on x naught marks where the car started.

For the whole run the car starts at zero and ends at three hundred four point eight meters, so its displacement is positive three hundred four point eight meters, and the positive sign says it points toward the finish. It works between any two points, too, so from the sixty-foot beam at eighteen point three meters to the three-hundred-thirty-foot beam at a hundred point six, the displacement is positive eighty-two point three meters, and all we needed were the two positions.

Displacements can be negative, too. Say that after the burnout the crew backs the car up to the line from fourteen meters out, then its displacement is zero minus fourteen, negative fourteen meters, and the negative sign just says it moved back toward the start.

Here are four things a race team might want to know about this run, and a single point can’t tell us one of them. Which one is it?

Reading the timeslip

Here’s the actual timeslip from that run. It looks like a receipt from a gas station, but almost every number on it is either a position or a time. The beams sit at sixty, three hundred thirty, six hundred sixty and a thousand feet, and since physics runs on meters, I’ll convert each one, so the finish line becomes three hundred four point eight meters. The only numbers that aren’t positions or times are these two speeds, and they deserve a closer look, because the track measured them with the same kind of beams that gave us all these times.

From positions and times, the first thing we can build is the average velocity, which is the displacement divided by the time it took, v average equals delta x over delta t. Velocity is a vector, so we write it with a little arrow on top, but on a straight track all of its direction lives in the sign, positive toward the finish and negative toward the start.

Let’s find the average velocity between the three-hundred-thirty and six-hundred-sixty-foot beams, and I’ll do it the careful way, starting with a quick sketch with an axis and a positive direction. From the slip, the car passes the first beam at a hundred point six meters and about two point one one seconds, and the second at two hundred one point two meters and about two point nine six seconds.

The definition uses exactly what we know, so we subtract, final minus initial, and get a displacement of a hundred point six meters over a time interval of zero point eight four two seconds. Dividing gives positive a hundred nineteen meters per second, about two hundred sixty-seven miles an hour.

Does that make sense? The sign is positive, so the car is heading toward the finish, and the size is a bit below the speed on the slip at six hundred sixty feet, which fits, because the car was still speeding up. Do that for every stretch and you get about twenty-two meters per second, then sixty-five, then a hundred nineteen, then a hundred forty-six, so stretch after stretch, the car keeps getting faster.

For the whole run, three hundred four point eight meters in three point six six three seconds gives an average velocity of eighty-three point two meters per second, about a hundred eighty-six miles an hour. But hold on. The slip says it was doing about a hundred fifty-five meters per second at the end of the run. Its average for the run is barely more than half of that.

Both numbers are right, because they answer different questions. The average only looks at where and when the run started and where and when it ended, so it knows nothing about how fast the car was going in between.

A lot of students try to get the average velocity by averaging the starting and ending speeds, which gives about seventy-eight meters per second here, and that’s wrong, because that shortcut only works when the acceleration is constant, and this car’s acceleration is anything but. Now you try one. Using the slip, find the car’s average velocity from the six-hundred-sixty-foot beam to the finish line, in meters per second.

The sixty-six-foot secret

So how did the track get that speed? A single timing beam only tells you the instant the car passes it, one position and one time, and that’s not enough for a speed, which needs a change in position and a change in time. So NHRA puts a second beam sixty-six feet before the finish line, about twenty meters, and times how long the car takes to get from one beam to the other.

On this run that took about thirteen hundredths of a second, and twenty point one meters divided by that is a hundred fifty-five point one meters per second, the three hundred forty-seven miles an hour on the slip. So the speed printed on the slip is really an average velocity. It’s taken over a stretch of track the car crosses in about an eighth of a second.

Let’s see why such a short stretch is good enough, by squeezing the two beams together around the finish line, from twenty meters, to two meters, to just twenty centimeters. It creeps up to about a hundred fifty-seven meters per second, and then it barely moves, because over a stretch that short the car can’t change its speed by much.

As we measure over shorter and shorter times, the average settles on one value, and that’s the big idea here, because an average velocity over a very small time interval is very close to the instantaneous velocity, the velocity at a single instant. The slip’s speed, a hundred fifty-five meters per second, is the average over the whole trap. Right at the line, the car is a touch faster, about a hundred fifty-seven, because it’s still speeding up. Two meters per second is close enough. From here on, we’ll use the slip’s number as the car’s speed at the finish line, three point six six three seconds in.

Now you run the timing system. Put two beams around the five-hundred-foot mark and squeeze them together until the average velocity stops changing, and that’s the car’s speed at five hundred feet.

How hard is it pulling?

Let’s go back to the instant the timing clock starts. The car is only just rolling, at a few meters per second, so what do you think its acceleration is at that same instant?

There’s almost no velocity arrow, because the car has only just started moving, but the acceleration arrow is enormous, and as time runs forward in slow motion, the velocity arrow keeps growing. So velocity and acceleration are two different things. Velocity tells you how fast you’re going and which way, and acceleration tells you how quickly that velocity is changing.

Just like velocity, acceleration has an average, and it’s the change in velocity divided by the time it took, a average equals delta v over delta t. Its units are meters per second divided by seconds, which we write as meters per second squared, but I like to hear it as meters per second, every second.

The track times the slip’s other speed the same way, over a short trap near the six-sixty beam. We’ll put each speed at the moment the car crossed that beam. At the start, almost zero. At two point nine five five seconds, a hundred thirty-four point seven meters per second. At three point six six three, a hundred fifty-five point one.

Over the whole run, the velocity goes from almost zero to a hundred fifty-five point one meters per second in three point six six three seconds. That’s an average acceleration of about forty-two meters per second squared. On average, the car added about forty-two meters per second to its velocity every second. A dropped rock gains only about nine point eight. That’s about four point three g.

Now it’s your turn to do one. Using the last two rows of that table, find the car’s average acceleration from the six-sixty beam to the finish, in meters per second squared.

On that last stretch, the car is faster than ever. Yet it’s pulling less hard, twenty-nine against forty-two for the whole run. The biggest velocity doesn’t come with the biggest acceleration. An object is accelerating whenever its velocity changes, in size or in direction, so even a car rounding a curve at a steady sixty miles an hour is accelerating, because its direction keeps changing. So speeding up, slowing down and turning all count, and the only way to have zero acceleration is to keep the same speed in the same direction.

Speeding up isn’t the only way to accelerate, either. Watch the arrows as he lifts off the throttle at the finish line, and again when the parachutes open. The velocity still points toward the finish, because the car is still flying down the track, but the acceleration has swung around to point backward, and that’s what slowing down looks like. Put three moments side by side and you can see how separate the two arrows really are, almost no velocity with a huge acceleration, then a long velocity with no acceleration at all, and then the two pointing opposite ways.

Now make a timeslip of your own. Launch it, keep your foot down all the way through the finish line, and the beams will print your numbers right next to Langdon’s.

Faster than sound, no engine

Now let’s go somewhere completely different. On October 14, 2012, Felix Baumgartner stood on a little step outside a capsule hanging from a balloon, thirty-nine kilometers above New Mexico, where the sky above him was black.

And then he leaned forward and let go. From the ground he’s just a speck, so the object model is easy, and he’s one point. Let’s put zero at the ground and call up positive, which makes his velocity negative as he falls. At the instant he steps off, his velocity is exactly zero, yet his acceleration is already about negative nine point seven meters per second squared, which is just gravity, a little weaker up there than at the ground.

With hardly any air up there to push back, he sped up almost as if he were falling in a vacuum. Thirty-four seconds after he let go, he passed the speed of sound, with no engine at all.

Fifty seconds after he stepped off, he was falling at three hundred seventy-seven point one meters per second, so with up as positive, what was his average acceleration over those fifty seconds, in meters per second squared? Notice that his acceleration is negative the whole time he’s speeding up. So a negative acceleration does not mean slowing down, and in a minute we’ll see exactly what it does mean.

His speed kept climbing, but not forever, so I’ve put two markers on his flight timeline for you. Slide them to either side of the moment you think his acceleration hits zero, then squeeze them together, and watch the average acceleration between them.

Look at the arrows now. The velocity arrow is the longest it will ever be, and the acceleration arrow is gone, because for that one instant his velocity isn’t changing at all. That’s the speed trap’s trick, the one you just used. Take his velocity two seconds before the peak and two seconds after, and the average acceleration between them is tiny, and as the window squeezes around fifty seconds, it heads straight for zero.

Now squeeze the same window around forty seconds instead. The change in velocity gets tiny there too, but the ratio settles near negative five meters per second squared, because forty seconds in he was still speeding up. Notice that the change in velocity shrinks in both windows, so a tiny change on its own proves nothing. What tells you the acceleration is the ratio, the change in velocity divided by the time, and that ratio goes to zero only at the peak.

Now think about what happens one second later. He’s still falling faster than sound, and his parachute is still packed away, so which way does his acceleration point?

So for the next few minutes he’s falling with his acceleration pointing up, slowing from more than three hundred seventy meters per second to under a hundred meters per second, and the parachute still hasn’t opened. Then, four minutes and twenty seconds after he let go, he opened his parachute, and even though he was still moving down, his acceleration suddenly pointed up, harder than at any other moment of the fall.

If we sketch his speed against time, it rises steeply, then more and more gently, and flattens out right at fifty seconds, then it falls as the thicker air slows him, and it plunges when the parachute opens, and his velocity, with up as positive, would be this same shape flipped below the axis. On the AP exam, a question like this comes with the word justify. A full-credit answer takes more than the right answer, so let me show you one.

Before I write the answer, which piece of evidence would you use to show that his acceleration was zero at fifty seconds? Here’s what I’d write. At fifty seconds, his acceleration was zero, that’s my claim, and I say it plainly. Then I back it up with what the graph shows, which is that his speed rose to three hundred seventy-seven point one meters per second at fifty seconds, peaked smoothly, and fell after that.

And then I connect that evidence to physics. Acceleration is how quickly velocity changes, and as we bring two moments closer and closer around that smooth peak, the change in velocity divided by the time between them gets closer and closer to zero, so at fifty seconds his acceleration is zero. An answer like, he wasn’t accelerating because he was at top speed, has the right idea, but it never says why top speed means zero acceleration, and that missing link is exactly what the reasoning has to supply.

What the sign of a tells you

Between them, our four real moments settle what the sign of the acceleration means, so let’s put all four in one table. At the launch, the velocity and the acceleration are both positive. So the car speeds up.

Under the parachutes the car is still moving forward, so the velocity stays positive, but the acceleration turns negative, and the car slows down. For Felix’s first fifty seconds, his velocity is negative and so is his acceleration, and he speeds up, which is exactly what that negative seven point five meant. After fifty seconds he’s still falling, so his velocity is still negative, but now his acceleration is positive, and he slows down.

So the sign of the acceleration on its own doesn’t tell you whether something speeds up or slows down. What matters is whether it matches the sign of the velocity, because matching signs speed it up, and opposite signs slow it down. That’s why the common idea that negative acceleration means slowing down is simply false, and Felix proved it for fifty seconds straight.

And if we had chosen down as positive for Felix, every sign in his column would flip. But both verdicts would stay exactly the same, because the physics doesn’t care which way you call positive.

Read the slip

Let’s finish by putting the whole lesson on one board. We’ll build it one line at a time. We can treat a moving object as a single point, keeping its mass, so it has one position at every instant. Displacement is the change in position, x minus x naught, and its sign tells you which way it points.

Average velocity is displacement divided by the time interval, and it only cares about where and when the motion started and ended. Average acceleration is the change in velocity divided by the time interval, and at any single instant, an object speeds up when its acceleration points the same way as its velocity, and slows down when they’re opposite. And if you make the interval tiny, like that sixty-six-foot trap, the average gets very close to the instantaneous value.

So the next time someone hands you a timeslip, you’ll know what every number on it measured, how to turn them into average velocities and accelerations, stretch by stretch, and why the shorter the stretch, the closer you get to what the car was doing at a single instant. And now you can point to any number on this slip. You can say exactly what the track measured on that run.