Instantaneous Speed: Jokers Wild

Last semester in help study, students were solving a problem to find out how fast an object would be moving right before it hit the ground, if it were dropped from 1 meter.

One student thought that it would be moving at 10 m/s, because 10 m/s/s is the acceleration due to gravity, and it had fallen 1 meter to get that speed.

A second student responded that it would take a whole second to reach that speed (not a whole meter). He added that it should take 10 meters to reach a speed of 10 m/s.

The Difficulty

I like both of these students' thinking, because it helps remind me of something: There are many conceptual pieces to acceleration that make it difficult to understand. One of those pieces is the concept of instantaneous velocity, and how it is different and related to constant and average velocity.

See, the first student is thinking that, in freefall, 10m/s of speed accumulates in 1m of distance. It doesn't. It accumulates over 1 second of time (no matter how much distance). The second student seems to get the accumulation of speed over time idea (at least in words), but is thinking of the 10 m/s accumulated as if it were the constant or average speed. While the ball will end with a instantaneous speed of 10m/s, it's average speed over the entire trip down will be 5 m/s, meaning it only covers 5 meters not 10.

Instantaneous Speed Taught:

Lot's of people teach instantaneous speed as a limiting process. You can do this with calculus, or with graphs (slope of tangent line). You can do this by zooming in on your calculator until it looks linear. You can do this by looking at really short segments of ticker tape. Whatever.

I think all of that is fine, but I think there's a better way:

I want to think about instantaneous speed as something like this: The speed you would measure if you could somehow keep the object from gaining or losing speed (long enough) to measure.

You don't NEED a limiting process to do this. All you need is a ramp.


If I want to measure the instantaneous speed of a ball somewhere along the ramp, I just stop the ball from speeding up or slowing down beyond that point. By making the ball go flat, I can measure it's constant speed, and that will be a good measure of how fast it was going right before it went flat.

If I want to profile the speed of the ball in many places along the ramp, I just repeat the procedure for various locations along the ramp. This has lots of extensions: you can change it up by starting it with some initial speed. You can ask students to predict how the final speed will compare to speed half way down. You can ask them to compare the d/t on the ramp to d/t on the flat part. Yah, yah, yah.

The Big Picture

I think there is real value to this. First of all, it is completely consistent with the concept of instantaneous speed. Instead of finding the slope of the tangent line on a graph; we are creating a tangent line with our measurement, by making the ball travel with constant speed–that's what a tangent line is. We determine the slope of that tangent line by actually measuring the distance and time of the ball along the flat path, instead of calculating a slope on a graph.

Second thing: This allows students to approach the idea of instantaneous speed without a limiting process. Now don't get me wrong. I'm going to teach my students to think about instantaneous speed as speed at an instant, and as a limiting process, and as a slope of a tangent line; but I'm going to do that when I need to. When will I need to? When I can't repeat an experiment over and over again. See with the ramp, I can send the ball down as many times as I want and "interrupt" the ball's ramp motion wherever I want. But what if I only have one shot to let the object move? Or what if I am watching a movie on you tube? Or how would you interrupt a ball falling vertically? All you get to do is observe, not experiment. I those cases, I'm going t have to work with my students to find a new method of measuring instantaneous speed.

Third, the "interrupt" method of measuring speed, especially with the ramp, has nice ways of engaging with the relationship among instantaneous speed (anywhere along the ramp), average speed (along the entire ramp), and the constant speed along any flat part. Building them as a set of relationships rather than separate concepts is necessary for students to understand the question this post started with.

I've only taught instantaneous speed this way once, but I thought it went quite well for giving students a way to think about the concept. I'm certainly not saying it's magic bullet, but it's certainly an approach worth consideration.

Kinematics: Acceleration is the Jack of Spades

One question I like to ask introductory physics students is this one:

Starting from rest, an object accelerates at 4 meters per second squared. What will be its speed in 3 seconds?

When I ask this question, I am looking for students to not look for an equation. I just want them to say "12 m/s, duh!" or, "Well, it would be going at 4 m/s after on second, and 8 m/s after two seconds, and 12 m/s after 3 seconds" or, "Well, every second that goes by, it gains 4 m/s of speed, so in three seconds it will gain 12 m/s."

Many students will say they have no clue without an equation and then give up, or they will go looking for an equation and then plug into the equation vf = at + vo. To me, either giving up or plugging into that equation tells me something important about that students' understanding of acceleration–they have no idea what accelerations means, at least not in a way that allows them to tell a story with it.

For the students who do succeed with or without an equation, I ask them how far it went during that time. To me, I think about this by saying that, on average, it must have been going with a speed of 6 m/s (half way between 0 and 12). Since that average speed happened over 3s, it must have gone 18m.

Nearly every student I encounter goes to the equation x = 1/2 a t^2. And while there's nothing wrong with that, I know these students are victims of the "Big 4", because I am fairly certain about two things:

99% of students who goes to an equation won't be able to solve both of these problems:

(a) In 4 seconds, a particle is accelerated from rest (by a constant force) to a speed of 20 m/s. How far has it gone?

(b) A bowling ball is hit impulsively with a rubber mallet, causing it to roll across the floor at increasingly faster speeds. If the ball is whacked with the hammer once per second, such that it speeds up by an addtional 10 m/s with each hit, how far will it have gone in the first 3 seconds after the initial hit.

The answers are:

(a) Over the 4 seconds, it's average speed was 10 m/s, meaning it went 40 m in those 4 seconds.

(b) In the 1st second, it travels 10m. In the 2nd second, it travels 20m. In the 3rd second, it travels 30m. In total, it travels 10m+20m+30m = 60m

Students Answers

In the first problem, students will likely say that the distance is 20 m/s * 4s =80m

In the second problem, students will likely say that x = 1/2 at^2 = 1/2 (10)*3^2 = 45m

How do you make sure your students understand acceleration?

I'm not sure. But I think the answer is to try to ask them questions that require them to make both sense of and tell stories that involve acceleration.

Kinematics: Is Average Velocity Queen?

In college physics, students often learn a slew of kinematic equations:

I'm a big fan of only teaching this one:


There are four reasons I like just using this, both personally and for teaching. It's all-purpose; conundrum-focused; tool-and-representation-infused; and intuition-refining.

First: The equation works all the time. There are no caveats to remember. As I've gotten older, this is the first of my two "go-to" ideas in kinematics. I will say that I almost never use the 2nd or 4th equation from above to solve any real or textbook problem.

Second: You get to engage students with the conundrums of figuring out what the average velocity is for a variety of situations: unchanging velocity, steadily changing velocity, piece-wise changing, non-linear changing, etc.

Third: You can explore various techniques like weighted averages or riemann-sums using data from tables, graphs, and other representations. It naturally builds in ideas of approximation.

Fourth: With these tools, you get to explore the boundaries of intuition (e.g., looking at this graph, what would you guess the average velocity to be? Or the famous example of determining the average speed when traveling 45 mph one way and 60 mph the other way on a round trip)

This is me guessing what the average velocity should be for a few situations:

At the end of the day, for instruction, you are likely to still have to fold in ideas about instantaneous velocity, changing velocity, and acceleration. It's something I want to talk about, but I'm going to leave it for another post.