# Distinctions: velocity at an instant, average velocity, and acceleration

**This week’s online pre-class question:**

An object is dropped from a height of 45m and takes 3 seconds to hit the ground. Explain why someone might think the object’s speed just before hitting the ground is 15 m./s. Then explain why that can’t be correct.

**Three responses representing very different places students can be:**

“First of all, wow! That’s the exact answer I had in mind and that is because if it’s dropped from a height of 45 meters and it takes 3 seconds to hit the ground you would want to divide the 45 meters by the 3 s to speed per second (15 m/s), but that is if it was going at a constant speed. So you also have to keep in mind that it was dropped at rest/zero so the speed will increase slowly not constant. I’m still confused.”

“Someone might think it is that because they would divide distance (45m) by time (3 s) which would come out to be 15 m/s. But that would be the average speed. To find the final speed you would take the initial speed (0 m/s) and add it to the acceleration (9.8 m/s^2) multiplied by the time (3 seconds). The final speed of the ball before it hits the ground would be 29.4 m/s.”

“Because most people would think just divided 45 into 3 to get 15m/s but we haven’t put in our minds about the acceleration of gravity, which is 9.8m/s that can round up to 10m/s then if you was to times 10m/s by 3s you would get 30m not 45m.”

Brian,

I love this question. Any plans to put together an archive of these questions for your fans?

I’m not sure I have enough for an archive, yet. However, I do think it helps to try to articulate what it is exactly you like about the question.

I’ll try to say what I like about it:

It puts student in contact with a common difficulty that many students will have–average vs. instantaneous velocity, and what acceleration means, etc.

It requires students to build the wrong reasoning themselves (rather than give them a fake student dialogue to react to), but it’s also likely that students will be able to do it.

It provides opportunity for learning at multiple-levels. From, “Wait, that’s not right?” to “Huh, I wonder why the final speed is double the average speed?”

It has a fairly low barrier to entry–there’s not a lot you have to know to begin thinking about it.

It has a flavor tending toward the meta-cognitive.

yep. You perfectly articulated why I like this question. It would be great to build a library of metacognitive questions like this.

I like it; I’ll have to steal it. Second the request for an archive.

But I also wonder what you think about the pros and cons of this alternate formulation:

An object is dropped from a height of 45m and takes 3 seconds to hit the ground. Another student states that the object’s speed just before hitting the ground is 15 m/s. Do you agree? If so, what is your reasoning? If not, why can’t this be correct?

Pros of your version are certainly that students are given the authority to decide whether it’s correct or not. For that reason, In class, I would certainly be inclined toward this version of the question. It sends the message that argument is a tool we use to settle correctness, it’s just not me saying so.

The negative, for me, is I think mostly an issue of timing with online quiz. In class, students would immediately have to contend with some saying it is correct, and some say it’s not correct. I can decide in class if and when to exert my authority, and to use that authority in ways that build off of and connect to student ideas.

In the online, quiz, if a student says, “Yes, it’s 15 m/s because 45m/3s = 15 seconds”… it sort of ends there. I suppose you could build in “answer dependent” follow up questions.

In the big picture, when I assign homework like this in my inquiry class, students have to do 3 parts:

(1) Give their answer and reasoning

(2) Discuss a wrong answer and what someone might have been thinking to arrive at that answer.

(3) Discuss the flaw in the wrong reasoning… (i.e., can’t just repeat what you wrote in #1).

These homeworks are writing intensive, and appropriate for my inquiry class. They have the advantage that, right or wrong, students have opportunity to engage with trying to construct an argument for the right answer, and also get practice sorting through and contending with counter-arguments.

Follow up thoughts: Thinking about the third students’ response, would a better question be with the numbers 20m and 2 seconds… explain why the speed is not 10 m/s.

My thinking is this: The student is thinking that as if “9.8” refers to a speed of gravity. He writes it in units of m/s, and his reasoning is consistent with this. He basically implies that 15 m/s is wrong, because it’s really 10 m/s.

My version of the question would make it harder for the student to fall back on that reasoning. Not sure this is better, of if it would be effective at all, but just a thought that crossed my mind.

I’m definitely stealing this question. I need more questions that get at whether my students are correctly distinguishing between velocity and acceleration…