Section 8.1
Door
My chair
My desk
If I start at my chair and walk to my desk and then back to my
chair, what is my displacement? Total distance traveled?
Displacement: 8+(-8)=0
Total distance: 8+8=16
Door
My chair
My desk
If I start at my chair and walk to my desk and then to the door,
what is my displacement? Total distance traveled?
Displacement: 8+(-10)=-2
Total distance: 8+10=18
What is the significance of a positive or negative displacement?
Is “total distance” ever negative?
f ( x)  position
f ( x)  instantaneous velocity
f ( x)  acceleration
Speed is the absolute value of velocity.
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?
200  200  200  100  700
700 feet
100
ft
min 50
0
200ft
200ft
2
4
6
8
10
minutes
-50
200ft
100ft
-100

What is the displacement of the bee?
200  200  200  100  100
100 feet towards the hive
100
ft
min 50
0
200ft
200ft
2
4
6
8
10
minutes
-50
-200ft
-100ft
-100

v (ft/sec)
4
14
t (sec)
7
5
10
-4
A turtle starts 2 feet to the left of my mailbox. Where is
the turtle at t=10?
What is the net distance traveled by the turtle?
What is the total distance traveled by the turtle?
v (m/sec)
4
2
2
4
1
8
4
t (sec)
5
10
-4
The velocity of a particle moving along the x-axis is given.
14  7  7
What is the displacement of the particle?
What is the total distance traveled?
14  7  21
If the particle started at x=3 when t=0, what is the position of the
particle at 10 seconds?
3  14  7  10
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.
Displacement   V  t  dt
b
a
To find distance traveled we have to use absolute value.
Distance Traveled   V  t  dt
b
a
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.

WITH CALCULATOR
An object moves along a coordinate line with the distance to the right
considered positive. If the velocity of the object is given by
v(t )  t 2  7t  10 over the time interval 1  t  3, find the net
and total distance traveled.
Set up the problem if you wanted to know the net distance for
0  t  7.
WITHOUT CALCULATOR
An object moves along a coordinate line with the distance to the right
considered positive. If the velocity of the object is given by
v(t )  t 2  7t  10 over the time interval 1  t  3, find the net
and total distance traveled.
Find the net and total distance traveled by the moving object in the
time interval given. Note that acceleration is given.
a(t )  2t  6, v(0)  8 ft / sec,from t  1 to t  5
The velocity of a car was read at 10 second intervals and recorded in
the table. Estimate the distance traveled by the car.
Time(sec)
0
10
20
30
40
50
Velocity
(ft/sec)
0
40
42
50
55
45
1
10  0  2(40)  2(42)  2(50)  2(55)  45
2
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