## Tuesday, April 12, 2011

### 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.

(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

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.

1. I like these examples, especially how you've tied it to the queen (v_avg for those who aren't following along ;)

I've posted this before but I really like g=22mph/s for the acceleration due to gravity. Of course it only works in backwards countries like the U.S. but, still, I think it gets to the essence of what you're posting here.

Keep up the good work, Brian, I'm really curious about the seven of hearts.

2. I also like g = 22mph/s. I just wrote a problem where students had to convert 10 m/s^2 into mph/s, and then figure out how fast a car would be moving if it were to accelerate at the same rate of g for 3 seconds.

I also like just talking about forces in terms of lbs.

I think the trouble with mph/s and lbs is when we get to Newton's laws-- we'd need to define some new mass unit as the mass that would accelerate at 1 mph/s with 1 lb force.