Why do a bowling ball and a car fall at exactly the same speed?

Today, we all know that any two objects, if dropped from a 20-story building, will fall at the exact same rate and hit the ground at the exact same time.  For many hundreds of years, though, this was considered counter-intuitive, and indeed, incorrect.  While society has been asking scientific questions about the movement of matter for over 500 years, an acceptable answer arrived much later than this.  A book by Isaac Newton published in 1687 explained the mathematical reasoning behind falling objects, and his theories stand today as the best models we have to predict gravitational acceleration.

So what does all that have to do with falling objects?  When I was younger, I was sure that a car and a bowling ball would fall at different speeds – after all, if the car is many times heavier, then it’s pushing downward at an equivalently high rate, and so it should fall faster.  This logic was widely held before Newton (and his predecessors) broke new ground in explaining the behavior of gravity.  Even if you know today that all objects fall at the same time, you might not know exactly why this is.  As it turns out, the answer is unbelievably simple.  It’s actually pretty amazing how obvious it seems in retrospect, but we have centuries of research to draw on, so whatever.

So why do all objects fall at the exact same speed?

The answer goes back to Newton’s second law, perhaps the most famous second law (with the possible exception of thermodynamics’s second law), which states that F = M * a. In English, the force acting on an object can be indirectly calculated by finding the object’s mass and also by looking at how fast it’s accelerating.  If we know these two things, we can come up with a numerical response to the “amount of bashing” that this object is experience.  You can push a car up a hill at a very slow rate, but if that rate is constant (say, one foot per minute), then this equation tells us that no forces are acting here.  This goes back to Newton’s first law – just because an object moves doesn’t mean it’s experiencing force.  When you’re pushing a car, your force exerted on the car is exactly cancelled by the resistance of the car to movement.  Sure, you have to exert some force to get the car going, but once it’s moving smoothly, there are no more forces at play.

So if we look at dropping objects off of a building, we can use this same principle: force equals mass times acceleration.  Since it’s not true that all objects have the same mass (does a car “weigh” the same as a pencil?), that means the 2 other variables in F = M*a must always equalize, no matter the object.  And indeed, this is the case.

First let’s look at our car.  It has a very high mass, and we know that acceleration is constant on earth, so that leaves only force, F, as the changing value here.  That’s no problem – for now, let’s say that a falling car has more force acting on it than a pencil.  At the same time, our pencil has much less mass, but still falls with the same rate of acceleration, so its force must also be much, much smaller than the car’s.

Here’s the final jigsaw puzzle piece here: the gravitational force between any two objects depends only on their masses.  It will change depending on the mass of each object, but no other factors matter.

Let’s drop our car off the side of a building once more, and say that it has ten times the mass of a motorcycle.  The car will have ten times the force acting on it as explained by F = M*a, but because the mass is ten times higher, the car is ten times more “resistant” to being moved.  Let’s see this in our equation:

Force on an object = its mass * its acceleration

F = M * a

Since we said our car is ten times heavier than a motorcycle, we’ll add a 10 in front of the mass.

F = 10 M * a

The force, then, is going to be 10 times higher than if the mass were equivalent to just 1 “unit.”

10 F = 10 M * a

Our 10s cancel here, and acceleration is STILL equal to F / M.  Repeat this calculation with a pencil, and you’ll see the same result: a = F / M. In other words, a car that’s ten times heavier than a motorcycle will experience ten times the forces, but this is only because its mass is ten times greater.  Since the force depends on the mass, we expect this to happen.  But because of the mass being ten times greater, the extra force pulling on the car is exactly cancelled by its high mass.

Stephen Hawking explains this perfectly in his book, A Brief History of Time.

One can now see why all bodies fall at the same rate: a body of twice the weight will have twice the force of gravity pulling it down, but it will also have twice the mass.  According to Newton’s second law, these two effects will exactly cancel each other, so the acceleration will be the same in all cases.

In fact, this mystery that eluded mankind for centuries is adequately explained by the simple equation we’ve used above.  There are no complicated numbers, constants, or imaginary numbers.  There’s no calculus, no slopes and no polynomial factoring.  The equation is so simple that it almost seems impossible, but you can use it just as we have here to explain exactly why everything falls at the same rate.  Kind of nuts, isn’t it?

Your car does calculus like a pro

You may have heard of the branch of math called “calculus.” To me, it always carried a connotation of “the hardest general math class in high school or university,” and after just a few weeks studying it, I’m not sure it’s all the horror it’s cracked up to be.  It’s a branch of math that lets us measure things that are changing as they change: if you drive 10 miles in 10 minutes, then your average speed was a mile per minute.  And if you drove 0 miles the first five minutes, then sped up to 2 miles per minute for the last 5 minutes, your average is still a mile a minute.  In our daily life, measurements like this really are taking the average of something – feet per second for a falling raindrop or degrees per hour, for example.  But most of the time, these things aren’t changing at a static rate — they’re changing quickly at some points and much more slowly at others.  Wouldn’t it be cool if we could measure these changes second-by-second to get a more accurate idea of how they change?

Calculus is nothing more than the study of change.  Specifically, we take a few concepts and apply them to the math that we already know.

The car example is not only one of the easiest to relate to, but it’s a perfect way to grasp just what calculus is all about.  For a car’s speed, you might use calculus to calculate the exact speed at exactly 4 minutes into a 10 minute trip.  Perhaps you’re moving at 1 mile an hour, or perhaps you’re not moving at all – calculus will show you your speed at any instant in time over the 10 minutes you drive.  But let’s show how we can take simpler math and upgrade it to super advanced by-the-second math.

What dose an odometer do?  It measures miles, of course.  If a drive takes you 45 minutes and you traveled 10 miles, then it took you 4.5 minutes per mile (you might be driving through New York City  if you’re driving this slowly).  All you have to do is mark your starting mileage and time, then when you’re done, you get your ending mileage and time.  Divide the change in miles by the change in time and you have your average speed.  This is algebra, or even simpler math: it’s something that most of us can do without breaking a sweat, perhaps because we actually use it quite often.  We might want to know how efficient the car is (miles per gallon), or how long it takes to get to a friend’s house.

So a car’s odometer measures average speed by keeping track of miles.  If you remember to record the time, you have the average speed.

So where’s the calculus?  Well, think about what your speedometer does.  You might argue it does the same as the odometer — with a little math, it gives you your mileage and time, but it’s really measuring your instantaneous speed directly.  If you go 25 MPH one minute then 35 MPH the next, you can look down at the speedometer during each of those minutes and see these numbers.  You can watch the needle go smoothly from 25 to 35 MPH, and you know with certainty that your speed changed during your trip.

Where an odometer measures averages, a speedometer measures instants — and that’s all calculus is.  It simply upgrades the concept of measuring averages to the concept of measuring instants of that same set of “stuff” that you’re measuring.

This graphic shows a curve that we can’t use regular math to solve – if this is our speed during the trip, we can use calculus to find out our exact rate of change of speed at one of these valleys.  You can find the speed by choosing an X value and following it to the corresponding Y value, but that only gives you a single instant’s rate.  Calculus will let you not only get this single point, but if you’re decelerating, it can give you the exact rate that you’re slowing down.  That’s something that can’t be done with traditional math.  If you’re accelerating, how in the world can you figure out by how much you’re speeding up.  That’s where the calc comes in, and it’s something your speedometer does constantly.

What’s really fascinating is that your car does this without a supercomputer – the needle and the speedometer it travels on are analog methods of solving a highly complex math problem, just like the hands of a clock.  Think about it this way: a clock will only show you the time down to the minute or second, but it doesn’t show you milliseconds.  You can estimate milliseconds from looking at the clock, but you can’t get them from the clock’s measurements.  The clock just isn’t that accurate.

So every time your speed changes, you’re really giving your car a calculus problem to solve.  You’re asking it to keep tabs on your speed as it changes second-by-second or millisecond-by-millisecond.  Your car does it all without griping, and the result is a little needle that jumps up and down the speedometer as you drive.

On a side note, Calculus can be used for many other cool things: if you can find the area of a square with algebra or geometry, then you can use Calculus to find the area of some many-sided shape your child drew at school.  Anything that’s changing so much that we can’t use algebra can likely be tempered with some calc.

I don’t know about you, but that’s just astounding to me.  Math is truly amazing sometimes!