Speed and Distance/Time Graphs

Turning Motion into Numbers

Once you know how to describe motion using a frame of reference, the next step is measuring it. Speed tells you how quickly an object's position is changing: specifically, how much distance it covers in a certain amount of time. If the truck in our crash scenario travels 60 meters in 3 seconds, its speed is 20 meters per second. That single number lets you compare the truck's motion to a bicycle, a sprinting cheetah, or another car, no matter how different those situations look.

Speed is calculated by dividing distance traveled by the time it took: speed = distance ÷ time. It sounds simple, but this one relationship is the foundation for predicting almost everything about how objects move, including how much time a driver has to react before hitting something.

There's an important detail hiding inside that simple formula, though: the speed you calculate this way is really an average speed over the whole trip, not necessarily the speed at any single instant. The truck almost certainly wasn't moving at exactly 20 meters per second the entire time. It might have been going a little faster during one second and a little slower during another. To know its exact speed at one specific moment, like the split second right before impact, you'd need its instantaneous speed, which is the speed at that precise instant rather than averaged over the whole trip. Speedometers in real vehicles are built to show instantaneous speed, updating constantly so a driver always knows how fast they're going right now, not just on average since they left the driveway.

Reading a Distance vs. Time Graph

Numbers are useful, but graphs let you see motion at a glance. A distance vs. time graph plots how far an object has traveled (on the vertical axis) against how much time has passed (on the horizontal axis). The steepness of the line, its slope, tells you the object's speed. A steep line means the object is covering a lot of distance in a short time, so it's moving fast. A flatter line means it's moving slowly, and a perfectly flat, horizontal line means the object has stopped completely.

If you saw a distance vs. time graph for our crash-test truck, you'd expect to see a fairly steep, straight line right up until the moment of impact. Then the line would go completely flat, because the truck's distance from its starting point stops increasing the instant it slams into the pole. That sudden change in slope, from steep to flat, is a graph's way of showing a sudden, dramatic stop.

You can actually calculate a precise speed straight off a graph using just two points on the line. Pick any two points, find how much the distance changed between them (the "rise") and how much time passed between them (the "run"), and divide rise by run. Suppose the graph shows the truck at 40 meters after 2 seconds, and at 100 meters after 5 seconds. The rise is 100 minus 40, or 60 meters, and the run is 5 minus 2, or 3 seconds. Dividing 60 meters by 3 seconds gives 20 meters per second: the exact same math as the basic speed formula, just read straight off a graph instead of a stopwatch.

Using Graphs to Predict a Crash

Here's why this matters beyond just reading charts: if you know an object's speed and its distance from an obstacle, you can predict when and how hard it will hit. Engineers and safety researchers actually do this kind of analysis using real recorded motion data. Tools like Vernier Graphical Analysis let scientists and students collect motion data from sensors (for example, tracking a toy car rolling toward a barrier) and instantly turn that data into a distance vs. time graph.

By studying the slope of that graph, you could estimate the truck's speed in the seconds before the crash and use it to predict how much force the impact would involve. This is exactly the kind of thinking real crash-safety engineers use to design safer cars, better seatbelts, and smarter warning systems.

What a Curved Line Tells You

So far, every graph in this chapter has shown perfectly straight lines, which only happens when an object moves at a constant, unchanging speed. But real motion is often messier than that. If the truck was speeding up in the moments before the crash, its distance vs. time graph wouldn't be a straight line at all. It would curve, getting steeper and steeper as time goes on. That's because the truck is covering more distance in each additional second than it did in the second before, which means its slope, and therefore its speed, keeps increasing.

A line that curves the other way, getting flatter and flatter, would mean the truck was slowing down, covering less distance in each additional second. Learning to recognize these curved shapes is just as important as reading straight lines, because most real-world motion (a car pulling away from a stoplight, a ball rolling down a hill, or yes, a truck losing control before a crash) involves speed that's constantly changing rather than staying perfectly steady.

Speed With a Direction Attached

Speed tells you how fast, but it never tells you which way. For a lot of real problems that missing half matters, so scientists use a second quantity called velocity: the speed of an object and the direction it is moving. If a car is traveling at 80 kilometers per hour toward the west, then 80 km/h is its speed and "80 km/h west" is its velocity. Velocity is often drawn as an arrow, where the length of the arrow shows how fast and the way it points shows which direction.

Here is the part students usually find surprising: an object's velocity can change even when its speed never changes at all. Imagine a car holding a steady 40 km/h as it drives north, reaches an intersection, turns left, and keeps right on going at 40 km/h. The speedometer reads exactly the same the whole time, so the speed is constant. But the velocity changed from 40 km/h north to 40 km/h west, because the direction changed. Velocity changes if the speed changes, if the direction changes, or if both change.

Two hikers can make the same point without anyone turning. If one hiker walks at 1.5 m/s south and another walks at 1.5 m/s west, they have identical speeds and completely different velocities, because they end up in very different places. Speed alone simply cannot tell those two hikers apart.

Notice how neatly this lines up with distance and displacement from the last chapter. Distance and speed are plain numbers with units. Displacement and velocity are numbers with units plus a direction. That is why a crash investigator cares about velocity rather than speed: knowing the truck was doing 20 meters per second is useful, but knowing it was doing 20 meters per second straight at a pole is what actually explains the wreck.