Sessions 2 and 3

Hi again, and welcome back to Physics 101! So nice to see you again. How've you been?

First, let us chat about order of magnitude questions, so-called "Fermi Questions" (as they were made legendary by Enrico Fermi). Here are a few we will discuss in class:

Number of days to walk around the world
Number of times your blink in your lifetime
Number of revolutions a car's tire will make
Number of dollar bills to stack to the Moon
Number of hairs on your head
Area needed for all people on Earth to gather for a party

We will discuss strategies for solving these questions today. Now clearly these are not physics questions, per se, but the way of thinking about them can be informative to our process. Physics is a way of thinking about problems, and often applying the language of mathematics to weird situations.

OK? OK!!!

And now......

THE EQUATIONS OF MOTION!

First, let's look at some definitions.

Vector vs. Scalar:

Vector quantities - quantities that must be expressed with a magnitude AND direction. Examples: velocity, displacement, force, momentum, magnetic field, electric field

Scalar quantities - quantities that only requires a magnitude (mass, time, energy, speed)

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Average velocity

v = x / t

That is, displacement divided by time.

Another way to compute average velocity:

v = (v0 + v) / 2

where vo ("v naught") is the original velocity, and v is the final (or current) velocity.

Average velocity should be distinguished from instantaneous velocity:

v(inst) = x / t, where t is a very, very, very tiny time interval.

[In calculus speak, this is: v(inst) = dx/dt, a derivative.]

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Acceleration, a

a = (change in velocity) / time

a = (v - vo) / t

The units here are m/s^2.

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Today we will chat about the equations of motion. There are 5 useful expressions that relate the variables in questions:

vo - "v naught", or original velocity. Note that the 0 is a subscript.
v - velocity at a specific point (usually at time t)
a - acceleration
t - time
x - displacement

Now these equations are a little tricky to come up with - we will derive them in class. (Remember, never drink and derive. But anyway....)

We start with 3 definitions, two of which are for average velocity:

v (avg) = x / t

v (avg) = (vo + v) / 2

and the definition of acceleration:

a = (change in v) / t or

a = (v - vo) / t

Through the miracle of algebra, these can be manipulated (details shown in class) to come up with:

v = v0 + at

x = 0.5 (vo + v) t

x = vo t + 0.5at^2

v^2 = vo^2 + 2ax

x = v t - 0.5at^2

Note that in each of the 5 equations, one main variable is absent. Each equation is true - indeed, they are the logical result of our definitions - however, each is not always helpful or relevant. The expression you use will depend on the situation.

Let's look at a sample problem:

Consider a car, starting from rest. It accelerates uniformly (meaning that the acceleration remains a constant value) at 1.5 m/s^2 for 7 seconds. Find the following:

- the speed of the car after 7 seconds
- how far the car has traveled after 7 seconds

Then, the driver applies the brakes and brings the car to a halt in 3 seconds. Find:

- the acceleration of the car in this time
- the distance that the car travels during this time

Finally, find:

- the total distance the car travels during this "experiment"
- graphs of motion for: displacement vs. time, velocity vs. time, acceleration vs. time

Got it? Hurray!

Physics - YEAH!!!