Increasing and Decreasing

I gave a presentation at AAPT on "Word Problems or Problems with Words" where I identified the difficulty of interpreting the words "increasing" and "decreasing," especially in the context of velocity and speed.

Technically, velocity is a vector and we can't talk about a vector as increasing or decreasing unless we specify that its magnitude increases or decreases. If the value of the x-velocity changes from -5 to 0, then we might say that it increased, but in this case we are thinking of the magnitude of the velocity (speed) and probably should use the word "speed" instead of "velocity." In this example, the speed decreases from -5 to zero.

However, the x-velocity can be negative or positive. Indeed, a velocity component that changes from a negative value to zero did increase. To better understand this, think of a number line. If you move to the right on a number line, the numbers are increasing (even if they are negative numbers). If you move to the left on a number line, the numbers are decreasing (even if they are negative numbers and their magnitudes are getting bigger). For example, the following numbers are increasing:

-5, -4, -3, -2, -1

The following numbers are decreasing:

-1, -2, -3, -4, 5

Suppose that this last list of numbers represents the x-velocity for a 1-D velocity vector. This is perhaps counterintuitive because the velocity component is decreasing yet the speed (magnitude) is increasing.

This confusion about negative numbers increasing as they get closer to zero creeps up again when you discuss potential energy. The ground state energy of H is -13.6 eV. If a H atom absorbs a photon, its energy INCREASES to, for example, -3.4 eV. Another example is the earth's orbit--as the earth gets closer to the sun, the gravitational potential energy of the system (which is negative) becomes more negative, meaning that it decreases! (This makes sense when you think of mgh, but since it's -GMm/r, a negative number increasing in magnitude, it's more confusing.)

I give my students the example of a checking account. If you have overdraft protection and your balance is -$250 and you deposit $100, did you balance increase or decrease? Did what you owe the bank (the magnitude of the balance) increase or decrease? They seem to identify with this, at least at the college level.