Motion - part 1

Motion!

THE EQUATIONS OF MOTION!

First, let's look at some definitions.

Average velocity

v = d / t

That is, displacement divided by time.

Another way to compute average velocity:

v = (vi + vf) / 2

where vi is the initial velocity, and vf is the final (or current) velocity.

Average velocity should be distinguished from instantaneous velocity (what you get from a speedometer):

v(inst) = d / t, where t is a very, very, very tiny time interval. There's more to be said about this sort of thing, and that's where calculus begins.

Now this idea (velocity) is pretty useful if you care about the velocity at a specific time OR the average velocity for a trip.

Also note:  if you travel at an average velocity for a period of time (say, 60 mph average, for an hour), the resulting distance is exactly the same as if you'd traveled at a constant velocity for the same time.

Some velocities to ponder....

Approximately....

Keep in mind that 1 m/s is approximately 2 miles/hour.

Your walking speed to class - 1-2 m/s
Running speed - 5-7 m/s
Car speed (highway) - 30 m/s
Professional baseball throwing speed - 45 m/s
Terminal velocity of skydiver - 55 m/s
Speed skiing - 60 m/s
Speed of sound (in air) - 340 m/s
Bullet speed (typical) - 900 m/s
Satellite speed (in orbit) - 6200 m/s
Escape velocity of Earth - 11,200 m/s
(That's around 7 miles per second, or 11.2 km/s)

Speed of light (in a vacuum) -

c = 299,792,458 m/s

This number is a physical constant, believed to be true everywhere in the universe. The letter c is used to represent the value being of constant celerity (speed).


On the other hand, if you care about the details of velocity, if and when it changes, then we need to introduce a new concept: acceleration.

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Acceleration, a

a = (final velocity - initial velocity) divided by time

OR

a = (change in velocity) / time

In equation form:

a = (vf - vi) / t

The units here are m/s^2, or m/s/s.

Acceleration is a measure of how quickly you change your speed - that is, it's a measure of 'change in speed' per time. Imagine if you got in a car and floored it, then could watch your speedometer. Imagine now that you get up to 10 miles/hr (MPH) after 1 second, 20 MPH by the 2nd second, 30 MPH by the 3rd second, and so on. This would give you an acceleration of:

10 MPH per second. That's not a super convenient unit, but you get the idea (I hope!).

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

vi - initial velocity. Note that the i is a subscript.
vf - velocity after some period of time
a - acceleration
t - time
d - displacement

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

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

v (avg) = d / t

v (avg) = (vi + vf) / 2

and the definition of acceleration:

a = (change in v) / t or

a = (vf - vi) / t

Through the miracle of algebra, these can be manipulated (details shown, if you like) to come up with:

vf = vi + at

d = 0.5 (vi + vf) t

d = vi t + 0.5 at^2

vf^2 = vi^2 + 2ad

d = vf t - 0.5 at^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.


In general, I find these most useful:

vf = vi + at

d = 0.5 (vi + vf) t

d = vi t + 0.5 at^2

By the way, note that the 2nd equation above is the SAME THING as saying distance equals average velocity [0.5 (vi + vf)] multiplied by time.

Also note that the 2nd and 3rd equations above both calculate d (distance).  However, in the 2nd equation, a is missing.  In the 3rd equation, vf is missing.  So, it's not as though one is more right than another - one may be more appropriate for the situation, depending on what you know in the problem.


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


Got it? Hurray!


There is another way to think about motion - graphically. That is, looking (pictorially) at how the position or velocity changes with time. We'll talk about this in class, and use a motion detector to "see" the motion a little better.

Physics - YAY!