OBJECTIVES:  

STUDENT HANDBOOK LESSON #4

I will recognize and use the basic equations of motion during today's class

WORDS O' TODAY:

• DISPLACEMENT (distance & direction)
• Distance
• VELOCITY (displacement/time)
• Speed (distance/time)
• ACCELERATION (velocity/time)

WORK O' THE DAY: 

How many sig figs?

1. 1.203 x 10^-15
2. 1000
3. 1000.
4. 1000.0

Derivatives Practice -- The following is a graph of the motion of an object in (how many?) dimensions (how do we know?) described the equation: x³-5x²

Where is the velocity of the graph 0? Why?

What is the velocity of the graph at 2.75 seconds?

Does the object have a velocity at -1.53 seconds? What does that mean?

Remember, the derivative is ALL ABOUT the rate of change of a graph at a specific, instantaneous point.

ANSWER:

So..... to find the value of instantaneous velocity of an object on a displacement vs time graph (can we do that on a speed/time graph?) we find the derivative of the equation of that graph and evaluate the derivative at a specific time.

The equation of the curve:

t³-5t²

Derivative of y with respect to x (the amount y changes for a given value of x):

3t²-10t

The slope of that curve at the instanteous point in time requested (t=2.75 seconds):

3(2.75)²-10(2.75) = -4.81 m/s

Which matches the orientation (negative) of the slope at that instant

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Now let's spend a few minutes talking about the basics of motion in one dimension.

For an object moving at constant motion in one direction the equation of motion are VERY basic:

• Velocity x Time = Displacement
  (V)(T) = D

The bad news is you will RARELY see situations this simple.

The frustrating news is that we tend to forget about this most basic of motion equations and I don't think it's even mentioned in our table of equations in the book

For an object moving in one direction at a changing velocity, it is OCCASIONALLY interesting to find the average velocity for that object if we don't really care about the velocity at any point:

• Average Velocity = (d_f - d_i)/(t_f - t_i)

Once again, this is fairly rare.

NOW FOR THE GOOD STUFF:

We are particularly interested in how an object behaves when the object is under constant acceleration in ONE DIMENSION:

Learn these, memorize 'em SOONEST. These are your new best friends:

1) v_f = v_i +at

2) v_avg = (v_i + v_f)/2

3) x_f = x_i + v_it + 1/2at²

4) v_f² - v_i² = 2ax

NOTES:

• v always denotes velocity
• a always denotes acceleration
• x means distance in the x direction, we will frequently swap this with y to mean distance in the y direction
• the subscript i always denotes initial
• the subscript f always denotes final

NOTES on working with motion:

1) Verify the object is moving in one dimension only (for now)

2) Determine whether gravity is involved:

• ALWAYS Yes in vertical motion situations... let's discuss
• ALWAYS No in horizontal motion problems... let's discuss

3) Change in displacement MUST result in velocity. Remember, displacement is a vector so it is possible to change direction without changing speed.

4) A change in velocity MUST cause acceleration. A positive acceleration resulsts in increasing velocity. A negative acceleration results in a decrease in velocity.