Preliminary

In the function

p(t) = p0 + vt

does the product vt mean

v added together t times, or

t added together v times?

Motion

II

Accelerated Motion

Finger tool• Position• Time• Position and Time• Speed• Speed and Time

How would you describe the motion of the cart?

Connect your calculator to the CBR with the 1/8” stereo jack cable:

Press [APPS] on the calculator and scroll down to choose CBL/CBR. At the main menu, choose Ranger. Then choose SETUP/SAMPLE. Use the arrow keys ◄►▲▼ and [ENTER] to change the options as below. You may use either meters or feet for the units.

MAIN MENU START NOW

REALTIME: YESTIME (S): 15DISPLAY: DIST

BEGIN ON: [ENTER]SMOOTHING: NONE

UNITS: METERS

Cursor up to “START NOW” and press [ENTER], then press [ENTER] again when prompted to begin the data collection.

Collect your own data:

Start your cart • At the top of the ramp or at the bottom• Moving fast or moving slow

Place the detector• At the top of the ramp or at the bottom• It is best to have the detector about ½ meter

away from the ramp since it does not detect close objects accurately.

Analyze your data in Fathom:

• A complete analysis of position vs. time (table, graph, formula with sliders, units)

• Compute the average velocity (V) between each data point

• A complete analysis of velocity vs. time(table, graph, formula with sliders, units)

• Compute the average acceleration (a) between each data point

• A complete analysis of acceleration vs. time(table, graph, formula with sliders, units)

Discussion

• What connections did you see in your three analyses: position, velocity, acceleration?

• Can you see these connections in other representations?

• What connections did you see across representations?

Average Velocity

• Formula:

• Graph:

• Context?

2 1

2 1

p t p t pv

t t t

time

posi

tion

p

t

Average Rate

Slope of the secant: The constant slope of an auxiliary line that gives the same amount of change in height (“rise”) over the same change horizontally (“run”).

Average velocity: The constant velocity for an auxiliary trip that gives the same change in position during the same time interval.

Average rate: The constant rate for an auxiliary situation that results in the same amounts of change in both quantities.