canceling terms

In my last post I talked about how I work with students with negative signs.  I thought I'd continue with some thoughts about canceling terms.  I'm a big proponent of having students solve problem algebraically first before plugging in numbers.  I usually reinforce that by asking follow up questions to problems like "what would happen if we increased the mass?", which are easier to answer if you can see whether it's in the numerator or denominator of the algebraic result.  On the way to an algebraic answer, though, there's many times when things cancel and I try to always be careful about those situations.

Here's an example: Consider a ball or block sliding down a frictionless ramp at the edge of a drop off.  The goal is to calculate the horizontal distance it will fly before hitting the ground.  Here's a quick sketch with the appropriate equations:

example problem with equations

Right at the point the particle leaves the ramp we can see that the mass cancels when considering the horizontal speed.  Right away with my students I would say "mass cancels!  That means you'd get the same answer with any block.  Cool!" or something like that.  The other major step in the problem is to figure out how long the flight will be.  Typically this is done by breaking the problem into horizontal and vertical components and finding when the vertical position has changed by the height of the drop off, as I've done on the left portion of the figure.  What's cool is that the final answer needs both major results (horizontal speed and time).  Both have "g" in them but together they cancel.  Here I'm often heard saying "g cancels! That means you'd get the same result on any planet.  Cool!" or something like that.

Doing the problem algebraically all the way before plugging in numbers lets students see what actually matters (in this case, the heights of the ramp and the drop off).  Students can then easily answer questions like "what would happen if you double h?" or "What would happen if you doubled the mass while tripling the strength of gravity?".  That last one causes groans for the students who've plugged their numbers in right away.

Another cool thing about this particular example is that the actual motion on different planets would be very different.  On the moon the horizontal speed would be very slow but the fall time would be long, exactly canceling each other!

When I push this method with the teachers that I teach (trying to get their physics license while already having another science license), most come to like the way that follow up questions are more fun and straightforward.  Some take a while to come around, though, as they like to grab their calculators right away.  The phrase "would it be the same on another planet?" usually works to get them to at least recognize the importance of canceled terms.

P.S. You get the same result for a ball that rolls without slipping on the ramp.  The horizontal speed decreases a little but mass still cancels (at least for spherically-symmetric balls) and the fall time stays the same.  "g" still goes away at the end.