# Reifying vector components

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 40^{o} 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 v_{0} 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 v_{0} and theta and they can use v_{0}*cos(theta) and v_{0}*sin(theta) to calculate x and y components of the initial velocity. A student can successfully complete part c) thinking of v_{0}*cos(theta) and v_{0}*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 v_{0}*cos(theta) and v_{0}*sin(theta) as entities that have physical meanings in and of themselves. By this I mean that a student who knows to use v_{0}*cos(theta) to calculate v_{0,x} does not necessarily know that v_{0}*cos(theta) can be used in equations in place of v_{0,x}. In order to solve part d), a student must see v_{0}*cos(theta) and v_{0}*sin(theta) as entities rather than just procedures. These expressions need to be seen as every bit as physical as v_{0,x} and v_{0,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=v_{0,x}*t and said ‘but I don’t know v_{o,x}‘. I asked if the student could relate v_{0,x} to the initial speed and the angle and he wrote v_{0,x}=v_{0}*cos(theta). I then pointed at this expression and said that we want to replace v_{o,x} in his first equation with v_{0}*cos(theta) because then we can solve for t in terms of v_{0}. The student’s response was, ‘Oh, I didn’t realize we could use this (v_{0}*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.