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Reifying vector components

March 9, 2014

I asked the following question when discussing 2D kinematics in Physics 1 this semester:

A ball is kicked with an initial speed of 32 m/s at an angle 40o above the horizontal.

a) What are the x and y components of the ball’s velocity 1 second and 3 seconds after being kicked?

b) How high will the ball go?

c) 60 m in front of where the ball is kicked is a 12 m tall wall. Will the ball make it over the wall?

d) For the same initial angle, what is the minimum initial speed required for the ball to make it over the wall?

Ignore the fact that 32 m/s is somewhat unrealistic as a speed for a kicked ball. What makes part d) more difficult than part c)? Take a moment to answer to yourself.

My initial thought when writing this question was that parts d) and c) differed primarily in their algebraic complexity. In part d), students need to solve for time in terms of v0 and then plug that algebraic expression for time into a second equation for vertical position. I was wrong. The algebra is nothing compared to the difference in how vector components need to be conceptualized.

In part c), students are given numerical values for v0 and theta and they can use v0*cos(theta) and v0*sin(theta) to calculate x and y components of the initial velocity. A student can successfully complete part c) thinking of v0*cos(theta) and v0*sin(theta) as nothing more than algorithms. A student who thinks of these expressions as procedures for computing x and y components with no physical meaning on their own can be successful on part c).

In part d), a student must recognize v0*cos(theta) and v0*sin(theta) as entities that have physical meanings in and of themselves. By this I mean that a student who knows to use v0*cos(theta) to calculate v0,x does not necessarily know that v0*cos(theta) can be used in equations in place of v0,x. In order to solve part d), a student must see v0*cos(theta) and v0*sin(theta) as entities rather than just procedures. These expressions need to be seen as every bit as physical as v0,x and v0,y.

This point was driven home to me when a student who had solved part c) asked for help on part d). I said (unhelpfully) that you follow the same process as for part c). The student wrote 60=v0,x*t and said ‘but I don’t know vo,x‘. I asked if the student could relate v0,x to the initial speed and the angle and he wrote v0,x=v0*cos(theta). I then pointed at this expression and said that we want to replace vo,x in his first equation with v0*cos(theta) because then we can solve for t in terms of v0. The student’s response was, ‘Oh, I didn’t realize we could use this (v0*cos(theta)) to, like, represent the x velocity.’ What an eye opener for me as an instructor!

This is a nice example of something Sfard calls reification. The Closes introduced me to Sfard and reification a while back, but I need to have another read and think about the implication for vector components.

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From → Ontology, Teaching

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