High school physics conceptual issue: why can't you use $v_f^2 = v_i^2 + 2ad$ to solve projectile motion x-component?

Question

My kid has a question in his homework that reads the following:

An astronaut stands on the edge of a lunar crater and throws a half-eaten Moon Pie horizontally with a velocity of $5.00\,\mathrm {m/s}$. The floor of the crater is $100\,\mathrm m$ below the astronaut. What horizontal distance will the Moon Pie travel before hitting the floor of the crater? (Remember that the acceleration due to gravity on the moon is only $1/6$ of that on earth)

It's been a while since I did physics, but I solved it quickly by finding the time using the formula

$$d_y = v_{i,y} + \frac12 at^2$$

Which gave me the time, then I used the same equation with the $x$ components to figure out $d_x$ (I got $60\,\mathrm m$ with sigfigs - seems to be right).

But my son asked why he can't use $$v_{f,x}^2 = v_{i,x}^2 + 2a_xd$$

since it seems to be a shortcut - we already know there's no acceleration in the $x$ direction since there's no wind resistance in these problems, and we know the initial (and therefore final) speeds in the $x$ direction as well?

But he got either a $0$ or an undefined depending on when he plugged in numbers, and I don't see what he obviously was doing wrong.

We just gave up, but it's been nagging at me ever since - would anyone happen to know?

Answer 1

I assume your undefined result comes from trying to solve your proposed solution equation for $d$

$$d=\frac{v_{f,x}^2-v_{i,x}^2}{a_x}$$

And since, as you have noted, $a_x=0$, this gives us an undefined value. But this makes sense! We can see this two ways.

First, if you don't solve the equation for $d$ we end up with $v_{f,x}=v_{i,x}$, which makes sense. There is no acceleration, so the velocity is not changing.

Second, it really does make sense that $d$ is undefined (technically "indeterminate", a better word for this discussion) because $a_x=0$ does not uniquely determine a traveled distance for set values of $v_{f,x}$ and $v_{i,x}$. You need to know the time of travel under constant velocity motion to uniquely determine the distance traveled. Unfortunately, this time has been eliminated from your proposed equation.

This is a great physics lesson for your child. Physics is more than just math. You need to think about the physical implications of what you are trying to do. If you get an undefined value it is because what you are trying to solve is actually undefined for the limited information you are trying to use. Also, just because an equation has a variable you need, it doesn't mean it is a valid equation. As I tell students I tutor, don't be "formula hunters" just looking for the right equation. Think through the physics first, then identify the equations you need that are valid for the system you are looking at.