Galileo and the two cyclists

Most people have heard of Galileo’s (alleged) famous experiment where two cannon balls – one heavy, one light – were dropped from the Leaning Tower of Pisa.  Which one hit the ground first?  Why not try the experiment yourself?  Find two rocks of different weights, and drop them at the same time from the same height (safety first!).  What do you observe?  The two objects hit the ground at the same time.  

There are two things you need to know to understand why this works.  The first is that bigger objects require more force to get them moving.  This is probably intuitive.  A baseball is easier to throw than a bowling ball.  The second physics principle at work here is gravity.  We all think of gravity as the force that pulls objects down to the ground.  Well, this force gets bigger for bigger objects.  So the heavier rock gets pulled to earth with more force.  But since it also took more force to get that big rock moving in the first place, it ends up falling no faster than the small rock.  Said another way, both rocks fall with the same acceleration – their speed increases at the same rate.

After learning that “all objects fall with the same acceleration”, an always curious and ever-incredulous student (the best kind) asked me, “so why does a heavier cyclist roll down a hill faster than a lighter one?”  It’s a good question, to which I couldn’t provide an immediate answer.  We just learned that gravity causes two objects, regardless of their mass, to fall at the same rate.  This should be true whether you are a rock falling straight towards earth, or a cyclist rolling down a hill.  So what gives?

Galileo wasn’t wrong.  It’s just that his model of falling objects was incomplete.  This is often the case in physics and all of science.  As Einstein once (maybe) said “Everything should be made as simple as possible, but no simpler.”  In other words, if the simple model works, great!  If not, we need to modify it in order to get a better description.  

Applying Galileo's model of the falling rocks to the two cyclists is too simple – it considers only the force due to gravity as affecting their downhill motion.  What else is pushing or pulling on the cyclist?  Well, at least so far, all cycling races take place on earth, where there is air.  This is mostly good of course: no cyclist could make it very far without air.  But air also creates air resistance, or drag.   So while gravity pulls the cyclists down the hill, the force of drag pushes back. 

The forces of gravity and drag compete when a cyclist rolls down a hill.

With Galileo’s experiment, we saw that the acceleration from gravity is mass-independent.  Not true for drag.  A given amount of drag force is less effective at slowing down a massive rider than a light one.  If you instead try Galileo’s experiment with a feather and a rock, you will see this principle quite clearly.  Drag has a huge impact on the feather (both because the feather’s mass is small, and the drag it experiences is large), causing it to drift slowly to the ground.   Taking gravity and drag into account, we can look at the speed of two riders over time as they descend down a hill.  You can see that the speed of the heavier rider increases more quickly and reaches a larger final value.

Speed vs time for two cyclists (150lbs and 200lbs) heading downhill.  Because of air resistance, the 150lb rider accelerates more slowly and reaches a lower terminal velocity.

This final value is known as terminal velocity – maybe you’ve heard the term before.  An object reaches terminal velocity when the accelerations from gravity and drag exactly cancel.  Zero acceleration doesn’t mean zero motion – it just means that your speed no longer changes.  A heavier rider has a larger terminal velocity.   

Get rid of the air and what would happen?  Apollo 15 astronauts performed exactly this experiment on the moon, where there is no air.  A hammer and a feather fall at the same rate when there is no drag!  So if cycling races took place on the moon, the heavier riders would have no advantage on the downhills.  Now accepting registrations for the Tour de Moon.