Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006

A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee? 200ft 100ft 700 feet

What is the displacement of the bee? 200ft -200ft 200ft -100ft 100 feet towards the hive

To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement. To find distance traveled we have to use absolute value. Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.) Or you can use your calculator to integrate the absolute value of the velocity function.

velocity graph position graph Displacement: Distance Traveled: Every AP exam I have seen has had at least one problem requiring students to interpret velocity and position graphs.

In the linear motion equation: V(t) is a function of time. (Differential equation) Separate. Integrate. Net distance - Displacement Integrate. Total distance.

a.When is the particle moving to the right, left, and stopped. b.Find the particle’s displacement if s(0) = 3. c.What is the particle’s final position? d.Find the total distance traveled by the particle.

a.When is the particle moving to the right, left, and stopped. Right: v(t) > 0; Left: v(t) < 0; Stopped: v(t) = 0

a.When is the particle moving to the right, left, and stopped. Right: v(t) > 0; Left: v(t) < 0; Stopped: v(t) = 0 The particle is moving to the right when 0 < t < Œ/3. The particle is moving to the left when Œ/3 < t < Œ/2. The particle is stopped when t = 0 and t = Œ/3.

b and c. Find the particle’s displacement and final position if s(0) = 3. The particle’s displacement is 2 meters and its final position 5 meters.

d. Find the total distance traveled by the particle.

The particle has traveled a total of 6 meters.

Important Idea Displacement = New position = old position plus displacement Displacement = accumulated rate of change

t =? sec. Try This Describe the linear motion : initial velocity(v 0 ) direction Speed (inc, dec, stopped?) position at t = 5 if s 0 = 1 ? ft/sec

t = ? sec. Solution 54 ft/sec right t = (0,3) then left t > 3 ? ft/sec Slowing down t = (0,3), t=3, speeding up at t > 3 Pos.=

Important Idea Distance traveled is related to displacement Distance = Displacement =

Try This What is the distance traveled by the particle during the first 5 seconds? Use GC w/ proper set-up. ft. in f/s

Summary The change in velocity is the accumulation of acceleration rates of change. The change in position (displacement) is the accumulation of velocity rates of change. These changes must be added to the beginning velocity or position to get new velocity or position.

Example From 1970 to 1980, the rate of potato consumption was millions of bushels per year with t being years since the beginning of How may bushels were consumed from the beginning of 1972 to the end of 1973?

Try This The average consumption rate in kwh of electricity for a home can be modeled by, where C(t) is measured in kilowatts and t is the number of hours past midnight. Find the average daily consumption in kwh.

Example A car moving with initial velocity, v 0, of 5 mph accelerates at the rate of 2.4t mph per second for 8 seconds. 1. How fast is the car going when the 8 seconds are up? What do you anticipate to possibly be an issue?

2. How far did the car travel?

Example Time min Rate gal/min The rate, P(t), at which a pump operates is recorded in 5 min. increments. How much water is pumped in 15 minutes? Hint: Trapezoid Rule

Assn. 48 pp : 1-7 odd, 12-17, 19, w/ work-justification and 37