If you are teaching or using physics, please DO NOT use this definition for the velocity in one dimension (it’s a bad idea):
“Velocity = distance divided by time.”
OK, technically there are some cases where this would be fine — but it leads to the really terrible version of velocity that looks like this:
This is really wrong. Oh sure, it can sometimes give you the correct answer — but even a broken clock is right twice a day. I mean I see why people use this equation. If they use x for distance and t for time then it’s just the same thing as distance divided by time. It also has the correct units for velocity. But it’s still wrong.
The Best Definition
Let’s get to it. Here is the best definition of the velocity in one dimension (for an object moving along the x-axis).
First, I will point out the “avg” part of this. This equation is for the average velocity in the x-direction. If you have an object with a changing velocity (like a car with a rocket pushing it forward) then the velocity is not constant — but this equation still works as long as you realize it’s the average velocity.
Second, the important part of this is the Greek letter, Δ (Delta). We use this to mean “change in”. So, the average x-velocity is defined as the change in position (Δx) divided by the change in time (Δt). It’s all about change.
If you look at the position and velocity at the beginning and end of some time interval, then we can call the starting position x1 and the final position x2. Doing the same thing for time, you can write the average velocity as:
What does that equation look like? Yup, that looks like the slope of a line. If you plot x vs. y (like you often do in math), then the slope is defined as:
Using a Position-Time Graph
Well, let’s do it. Let’s make a graph. Here is a very simple position-vs-time graph for a car moving along with a constant velocity.
What is the velocity of this car? Suppose I want to use the wrong definition of velocity — x/t. I’m going to pick the point at t = 1 second (and x = 0.8). This would give a velocity of:
Just to be clear, this is wrong. What if you pick a different time? Let’s say t = 5 seconds. OH! Look! The x-position is 0 meters at that point. That would give a “wrong” velocity of 0.04 m/s.
No. The velocity is defined as the change in position divided by the change in time. That’s the slope of the x-t graph. As you can see above, the line is indeed straight. This means that I can pick any two points to find the velocity. I will just pick some that are easy to find. How about these:
Using these values, I get an average velocity of:
It’s always about change.
More Complicated Motions
What if the car had a position-time graph like this:
What now? How do you find the velocity? Well, the first option is to just find the average velocity. What about the average velocity from time t = 2 seconds to t = 4 seconds? Yes, we are going to have to get an estimate of the position from the graph. How about x1 = 2.2 meters and x2 = 3.4 seconds?
Using those two values, I get an average velocity of 0.6 m/s. But what if I use the positions at t = 3 seconds and t = 5 seconds? That would give an average velocity of -0.295 m/s.
Both of these values are correct. Both are average velocities. It’s just the velocity of this car is changing. It’s not constant. What if I want the velocity of the car exactly at a time of 4.5 seconds? I could pick times of 4.0 seconds and 5.0 seconds. This would give an average velocity of -0.91 m/s. But that’s not quite right.
In fact, the smaller the time interval the better the approximation to the actual velocity at that point. This graph isn’t perfect because you can’t see the big difference in slopes, but they are indeed different.
The slope of the tangent line (and the actual velocity at 4.5 seconds) is -0.875 m/s. But the key point I’m trying to make here is that it’s NOT x divided by t.
Please don’t use that equation in physics. Velocity is all about a rate of change (change in position divided by a change in time).