This section is for those students who don’t remember (or perhaps never were taught) elementary physics. I hope to give some qualitative notions of some basic concepts in physics and to do so with a minimum of mathematics, using pictures, animations and links to available explanations on the web. So, dear reader, imagine you’re living in pre-Renaissance Europe, and are listening to those Medieval monks explain what they think about motion, and how it differs from what Aristotle had to say.
5.2 DISTANCE, VELOCITY, ACCELERATION
First, let’s consider distance. I believe all you readers have an intuitive notion of what distance is: you draw a straight line between point A and point B and the length of that line is the distance between points A and B.¹
What is velocity, then? Velocity is a rate, distance per time. (And, to be fussy, velocity has direction; “speed” is the magnitude of velocity; you don’t care what the direction is; velocity is “speed” plus direction.)
Now I ask your pardon, dear reader to bear with me while I inject just a little math to make the concept clear. Suppose it’s four miles to the nearest rest stop on the thruway and you must get there in five minutes (or less–I won’t ask why. How fast do you have to travel or what should your car’s velocity be? Your rate of travel, speed, must be four miles in five minutes, or 4miles/ 5 minutes, or as it would be written conventionally, 4/5 miles/minute; in other words, distance divided by time. Since there are 60 minutes in an hour, a little arithmetic shows you would have to travel 60x (4/5) miles/hour or 48 mph². And here’s an equation (again, pardon)
v= d/t where v is velocity, d is distance and t is time to travel that distance
What is acceleration? It’s also a rate, the change in velocity divided by the corresponding change in time. Let’s turn again to an example with some numbers. Fresh out of grad school I bought a MG TD (red, no less!). The MG was not, to use my grandson’s lingo, “zippy.” From a standing start, it could get to a speed of 42 mph in about 20 seconds (real sports cars take only about 5 seconds to get to 60 mph). This acceleration rate corresponds happily (for nice numbers) to about 1 (m/s)/s or 1 m/s². So we have acceleration, a, given by the gain in velocity over the time, t, it takes to achieve that change:
a = change in v / t
The animation below shows the MG going from standing start to final velocity, 42 mph, in 20 seconds and traveling thereafter at constant velocity of 42 mph. The shots are taken at 4 second intervals starting from 8 s after the start.
You can use the play controls, play and pause, at the side of the animation to stop and start the animation; also you can select a button at the bottom to show where the MG is at given times: 8, 12, 16, 20, 24, 28 seconds after start.
Also shown below is a velocity profile at the times after start; the speed is listed above each car image; the arrow length corresponds (roughly) to the velocity:
An easy way to think about constant acceleration is that the distance covered in a given time is average velocity multiplied by the time. The average velocity is just (1/2) (v_beginning + v_end).³
As pointed out in SECTION 3.2, Nicolas Oresme had derived these relations between velocity, distance and acceleration by a graphical analysis, 100 years before Galileo. However, it was Galileo who did the science: confirmed the theory by experiment.
How did Galileo set up an experiment where the motion would be slow enough for him to measure time, distances and speed? Acceleration of falling bodies would be too fast. Here’s the experiment, done in elementary physics lab classes. An inclined plane, as in the illustration below, length L, is set up so that the top end of the plane is a height h above the ground. A ball or cylinder rolls down the plane and you measure distance traveled in given times. Now if the plane were to be vertical (h=L), the ball would fall with an acceleration that of gravity (9.8 m/s²) and that would be too fast. If the plane is flat (h=0), the ball would not roll at all (hey! that’s poetry?). Clearly the acceleration is going to vary as the height h changes. It turns out that the acceleration is proportional to h/L. It will be the same–independent of size or material–for a given shape sliding or rolling down the plane.
How do objects acquire velocity, that is accelerate? Buridan in the 14th Century had ideas about velocity that anticipated Galileo and Newton centuries later. He said that a moving body had “impetus,” the heavier the body moving at a given velocity, the more impetus it had. If you threw a ball, the motion of your arm gave the ball its impetus. “Impetus” is what we now call “momentum” and define as
momentum = mass x velocity
Mass is what we ordinarily think of as weight, but to be fussy, weight is really mass times the force of gravity. You can think of mass as resistance to change in motion, what would technically be termed “inertia.”
Here’s an example to give you some intuitive notion about momentum: the MG TD referred to above is a very light car, weighing only about 1/2 ton (1000 pounds); a late model Cadillac is much heavier, weighing about two tons. Accordingly, the mass of the Caddy is about four times greater than that of the MG. So, if the MG were traveling at 40 mph and the Caddy at (1/4)x 40 mph = 10 mph, they would have equal momentum (if they were traveling in the same direction–remember, velocity has direction, speed does not). This is illustrated in the animation below.
What causes a body to accelerate, acquire velocity? Again, Buridan had the right qualitative notion: the body acquired impetus because of an action by an agent, you, throwing the ball with your arm. In this notion there is an implied notion of force, which Newton (17th century) made explicit by his Second Law of Motion:
Force = mass x acceleration
more generally if mass doesn’t stay constant (think of an example involving liquids!)
Force = change of momentum/change of time
For the first definition, go back to the example of the accelerating MG: the force is provided by friction between the tires and the road, the tires—wheels—are made to go round by the engine turning a drive-shaft.
For the second definition, think of a pitcher winding up and releasing a baseball moving at 90 mph as depicted in this video . The baseball has a mass of about 0.15 kg (or about 0.3 lbs) If you go frame by frame in the video, you’ll see that it takes less than 10 ms (0.01 s) for the pitcher to start his windup and release the ball; that’s the change in time for the baseball to acquire its velocity of 90 mph (we’ll neglect air friction slowing the ball down). So, fussing with units—I don’t need for you all to mess with the arithmetic—you get a force of about 650 Newtons required.
For comparison, the force of gravity on the baseball is about 1.5 Newtons. If air friction is neglected, from what height would the ball have to fall to get this 90 mph velocity? About 100 yards. Why the greater force to throw the ball this fast? Because the force of the throw is acting for only a short period of time, during the pitcher’s windup, whereas gravity will be acting all during the fall.
5.5 ENERGY AND WORK; CONSERVATION OF ENERGY
There are two other physics concepts, as important as velocity, acceleration and force, that bear on motion, and those are energy and work. To get an intuitive idea of this, let’s go into more detail about how the MG acquires velocity.
First, fuel is burnt in the cylinders to move the pistons up and down and thereby rotate the shaft that turns the rear wheels around, moving them against the friction of the road. We have then chemical energy from the gasoline combining with oxygen (burning) converted to mechanical energy. The energy of motion is called “kinetic energy” and is given by the formula
Kinetic Energy = (1/2) mass x velocity^2
Another important form of energy is “potential energy,” energy a body has by virtue of its position. Let’s think about what this means. When you let a ball roll down an inclined plane it has zero kinetic energy at the top and kinetic energy at the bottom after it’s accelerated due to gravity and acquired velocity. So where does that kinetic energy come from? To balance the energy books we say the ball at the top of the plane has potential energy that can be converted to kinetic energy. This potential energy is given (for gravity at the surface of the earth) by
Potential Energy = mass x g x h= mgh
where g is the acceleration due to gravity (9.8 m/s^2), h is the height above the bottom
This is illustrated below:
An important principle of physics is that energy is conserved. What does that mean? It means that energy doesn’t disappear into nowhere, for example:
- kinetic energy, energy of motion is lost due to friction, but is converted to the same amount of heat energy;
- kinetic energy, energy of motion is lost due to work done, moving the MG up a hill–the work done is equal to the amount of kinetic energy lost; the work done is equal to the gain in potential energy at the end;
- chemical energy of the gasoline is converted to kinetic energy less friction losses in the engine and drive shaft.
Accordingly, the energy bank account balances: input (at the beginning) of chemical energy, gasoline in the fuel tank = kinetic energy at the end of the drive, when the fuel tank is empty + energy lost due to friction of the tires with the road, engine and drive shaft friction + work done due to a net change in height level at the end or gain in potential energy.
In Essay 2, we’ll have more to say about energy and the science of energy, “Thermodynamics,” particularly those two important laws: The First and Second Laws of Thermodynamics.
¹Let me add a cautionary note physicswise: if you are traveling between A and B (home and the local fastfood place, let’s say) and you wander around, make side-trips, the distance is still the length of the line between beginning and ending points. If you want to get total mileage traveled, then you have to draw straight lines between each of the intermediate starting and stopping points and add the lengths up.
² Since each hour contains 60 minutes, you would have to go 60 (
minutes/hour) x (4/5) (miles/ minute) or 60 x (4/5) (miles/hour)= 48 (miles/hour).
³For our example, the distance covered by the accelerating MG between 12 seconds (v_beginning = 25 mph) and 16 seconds (v_end=33mph) is just
(1/2) (25+33) (miles/
hour) x(1 hour/ (3600 seconds) x (16-12) seconds or about 56 yards