Class Notes, Sep 11

Class Notes for Fall, 2000

Week of	Aug 23	Sep 6	Oct 2	Nov 6	Phy 205 page
	Aug 28	Sep 11	Oct 9	Nov 13
		Sep 18	Oct 16	Nov 20	Notes for Fall, 1999
		Sep 25	Oct 23	Nov 27
			Oct 30

Phy 205, Week 4

11 Sep The two key equations in chapter two are

v_x = dx
dt
and a_x = dv_x
dt

The precise definitions of these derivatives as limits is on the preceeding page.

To see how to use them, start with an example and walk through the definitions. Take

x( t ) = A t² + B t + C

A, B, and C are three constants, and I picked particular numerical values in class. Apply the definition of velocity to this and you get

v_x =
lim
Dt Ć 0
x( t + Dt ) - x( t )
Dt
= dx
dt

For this example,

x( t + Dt ) - x( t ) = A ( t + Dt )² + B ( t + Dt ) + C - A t² - B t - C.

When you expand and simplify this, you get

2 A t + A Dt + B.

Finally, take the limit as Dt approaches zero and the middle term goes away, leaving the function

v_x( t ) = 2 A t + B.

A similar set of calculations for the acceleration, gives

v_x( t + Dt ) - v_x( t ) = 2 A Dt.

When you divide this by Dt, you get 2A (even before taking a limit). In this special case, the function a_x(t) happens to be a constant.

In most cases you don't know the position of an object as a function of time; that may be what you are trying to find out! The problem is usually the reverse of what I've just done. Instead of finding the derivative of a function, it is often necessary to find the anti-derivative.

The anti-derivative of a function is the function whose derivative is what you started with.

The simplest common example of this is gravitational free fall. If the only force on a object is from gravity, then the acceleration (near the Earth's surface) is constant. I worked out the example of throwing a baseball straight up at an initial speed of say 50 mph = v₀. The experimental observation about gravity near the Earth is that the acceleration is constant. Choose the x-axis to be positive upward, then the equation to solve is

a_x = dv_x
dt
= - g

Here "g" is the magnitude of the gravitational field, = 9.8 m/s².

I worked backwards from this to find the velocity and the position as functions of time. There are two arbitrary constants that show up in this calculation. These are determined from the initial conditions stated in the problem. I threw the ball up with a given initial speed and I will select my coordinate system so that the origin is where I threw it from.

The anti-derivative of the equation dv_x/dt=-g is

v_x( t ) = - gt + C,

where C is some unknown constant. The next step is to take another anti-deriviative, because the preceeding line is dx/dt:

dx/dt = - gt + C implies x( t ) = - gt²/2 + Ct + D,

where D is still another unknown constant.

To evaluate these two unknowns requires two equations. These come from the unused part of the information in the statement of the problem.

At t = 0, x = 0 and v_x = v₀.

I have some freedom here. I chose the coordinate system so that the origin is at the position from which I throw the ball. The two equations are now

v_x(0) = - g 0 + C = v₀ and x( 0 ) = - g 0²/2 + C 0 + D = 0.

These two equations determine C = v₀ and D = 0.

Put the equations together with the initial conditions and you have

x( t ) = - g t²/2 + v₀ t and v_x( t ) = - g t + v₀.

The future motion of the ball is now known, at least until it hits the ground.

13 Sep Discuss in some detail a homework problem that involves relative velocities. The main difficulty is in picturing what is happening. I drew some pictures that may help. Also, it helps to view the problem from the perspective of the person who is moving.

Recapitulate the problem from Monday, cutting through to the essential steps. Then do another example the same way: From the old exam, the airplane takes off, reaching its takeoff speed of v in a distance L. As before, start with
1. what is given
2. a crude sketch
3. pick a coordinate system
4. state all the given data in terms of the coordinate system; give names to the unknowns
5. use the basic definitions of acceleration and velocity, taking anti-derivatives
6. evaluate the constants from the initial data
7. use the rest of the information given to find equations for the other two unknowns
8. try some numbers to see if the result is plausible.

15 Sep Do in detail several example problems. One from an old exam, two more from today's homework.

Solve all these problems the same way.

Start with a sketch and what is given. Use the basic equations

v_x = dx / dt,

a_x = dv_x / dt

Whether the acceleration is a constant or not, start here and find the anti-derivatives. Use the initial conditions to determine what the constants are, and you may get into some messy algebra, but the concepts are always the same.

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