Little Bunny Foo Foo’s Guide to Phyics – Pt 1: Inertial Reference Frames
One day, Little Bunny Foo Foo was hopping through the forest at a constant speed in a specific direction. As everyone knows, speed is a measure of how fast something is going. When we care about the direction as well as the speed, we use the word “velocity”. Velocity (usually written as a v or u with an arrow on top) is a vector, having both magnitude (how fast) and direction (where to). Just like velocity is a measurement of the change in position over a period of time, acceleration is a measurement of the change in velocity.
Because Little Bunny Foo Foo is hopping with constant speed as well as constant direction, he is not undergoing any acceleration. This is important because, in the 1600’s, a guy named Newton showed that the sum of all forces exerted on an object is equal to the object’s mass (m) times the object’s acceleration (a. When you can’t use an arrow above the letter to denote a vector, you type it in bold). This is known as Newton’s Second Law of Motion (F = ma)
Newton made many of his most famous mathematical and physical discoveries during a hiatus from university. He spent time in the country avoiding the Black Plague, which was ravaging London at the time thanks to very healthy rat and flea populations.
Foo Foo himself, who lived during the same time period as Newton, had lost 32,000 of his nearest relations to the plague and was actively seeking revenge on the local rodent population. This is unfortunate for the field mice, who were summarily plucked off the ground by an enraged bunny and decapitated. (Most historical accounts claim that the field mice were bopped on the head, but this is simply a poor translation.)
This all explains why the field mice were also running through the forest in a specific direction (away from Little Bunny Foo Foo) and with a constant speed, which was as fast as they could go. Sadly, it just wasn’t fast enough. The field mice were only able to run about two thirds as fast as Little Bunny Foo Foo. But, because the field mice were also moving with constant velocity, the sum of all the forces acting on the them also added up to zero.
Who cares, right? That Newton guy again, for one. Newton’s First Law of Motion is often called the Law of Inertia. The most common version of this law states: an object in motion tends to stay in motion and an object at rest tends to stay at rest unless acted on by an external force. This is almost, but not quite, correct
Imagine you’re on a bus traveling at a constant speed. You set an apple on the empty seat next to you. What happens? Well, nothing. The apple stays right where you put it. Suddenly, though, the bus driver slams on the breaks. The apple flies forward and hits the back of the seat in the forward row. You know, of course, that the apple flew out of the seat because the bus suddenly accelerated in the direction opposite of its motion (people call this decelerating usually, but physicists are very precise).
Now imagine that the bus’s engine made no sound or vibration and that all of the windows were blocked. When the bus is traveling at a constant speed, everything stays put. Suddenly, an apple on the seat next to you flies forward for no reason (with no sensory input to tell you different, you don’t know the bus has suddenly decelerated). Because the apple appears to be experiencing acceleration, it looks as if an external force must have flung the apple out of the seat. This isn’t exactly true. The apple didn’t speed up. The bus slowed down.
This is why scientists have to be very careful, not only about how they take their measurements, but about the frame of reference in which those measurements are taken. Someone measuring the position of trees from a car accelerating on the freeway might erroneously come to the conclusion that the trees are accelerating at the same rate, but in the opposite direction. This person would, of course, be an idiot. Still, sometimes its not so apparent exactly what is doing the moving relative to whom.
The Law of Inertia is more correctly described this way: an object in motion remains in motion and an object at rest remains at rest, unless acted on by an external force and when viewed from an inertial reference frame. An inertial reference frame is simply a reference frame that isn’t undergoing any acceleration. It’s either moving at a constant speed, or it’s at rest.
In other words, before saying anything about the apple’s acceleration, you have to first know what your own is. The easiest way to measure the acceleration of something else is when your own acceleration is zero. What’s even better is that the law of physics are exactly the same in every inertial reference frame, and any inertial reference frame is as good as the next, no matter what its speed or direction. There is no absolute frame of reference. Everything in the Universe is accelerating away from everything else, all the time, speaking on a stellar level. So measurements of acceleration are always taken relative to something else. Because it makes the math so much easier, inertial reference frames are always preferred (even though the frames themselves are only moving at constant velocities relative to other things).
This explains how, when a group of physicist ticks traveling on the back of Little Bunny Foo Foo at a constant velocity of six kilometers per hour due East, measured the acceleration of the Good Fairy, their results agreed with data gathered by a similar group of physicist fleas traveling on the back of the field mice, who were themselves moving at four kilometers per hour in the same direction.
The good fairy was accelerating at a constant rate, which both the ticks and fleas appreciated as they didn’t have to do any calculus. When the ticks started timing the Good Fairy’s progress, they clocked her at 2 km per hour, and when they took another reading five seconds later, she was approaching at a steady clip of 17 km per hour. To calculate her acceleration, they took the difference between the two speeds and divided by the time interval between measurements: (17 – 2)/5 = 3.
The good fairy’s acceleration, according to the ticks on the back of Little Bunny Foo Foo, was 3 kilometers per hour squared. That meant that, for every hour the Good Fairy accelerated, her speed would change by three kilometers per hour.
The fleas on the back of the field mice took measurements of the Good Fairy’s velocity over the exact same time interval as the ticks. Because they were moving two kilometers an hour slower than the ticks, they initially measured the Good Fairy’s approach at 4 km/h and her final speed as 19 km/h. (19-4)/5 = 3 km/h.
Had the fleas and the ticks not been in inertial reference frames themselves, their measurements wouldn’t have agreed. Years later, Albert Einstein ran with the notion that all the laws of physics really were the same in any inertial reference frame, including the constant speed of light through a vacuum, and formulated his Special Theory of Relativity.
Fortunately for the field mice, the Good Fairy’s acceleration relative to both mouse and bunny was in the positive direction, and many a cute, furry mouse was saved from a grim fate.