How do We Measure Speed (2.1)
Velocity Versus Speed
Definition 1 Scalars are quantities that are fully described by a magnitude
(or numerical value) alone.
Definition 2 Vectors are quantities that are fully described by both a magnitude and a direction.
Definition 3 Speed is a scalar quantity that refers to how fast an object is
moving.
Definition 4 Velocity is a vector quantity that refers to the rate at which
an object changes its position.
Example 1 An example of a speed is 55mph. An example of a velocity is
55mph North.
Average Velocity
Definition 5 If s(t) is the position of an object at time t, then the average
velocity of the object over the interval a ≤ t ≤ b is given by
Average Velocity =
∆s
∆t
=
s(b)−s(a)
b−a
In words, the average velocity of an object over an interval is the net
change in position during the interval divided by the change in time.
Example 2 The trajectory of a grapefruit
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Table: Height of the
t (sec)
0
1
2
3
4
5
6
Grapefruit vs Time
y (feet)
6
90
142
162
150
106
30
1. Compute the average velocity of the grapefruit over the interval 1 ≤ t ≤
3. What is the significance of the sign of your answer?
2. Compute the average velocity of the grapefruit over the interval 4 ≤ t ≤
5. What is the significance of the sign of your answer?
Note 1 In calculus we use the following notation:
Average Velocity =
s(a+h)−s(a)
h
Example 3 Graph: Height of the Grapefruit vs Time
2
Note 2 The average velocity over any time interval a ≤ t ≤ b is the slope
of the line joining the points on the graph of s(t) corresponding to t = a and
t = b.
Example 4 Calculate the Average Velocity, over the time interval from 1
second to 2 seconds, of the grapefruit whose trajectory is given by the graph
above.
Solution.
average velocity =
s(2)−s(1)
2−1
average velocity =
=
= 52 fst
142−90
1
s(1+1)−s(1)
1
=
142−90
1
= 52 fst
Problem 1 The distance, s, a car has traveled on a trip is shown in the
table as a function of the time, t, since the trip started. Find the average
velocity between t = 2 and t = 5.
t (hrs)
0
1
2
3
4
5
s (kms)
0
45
135
220
300
400
Problem 2 The table gives the position of a particle moving along the xaxis as a function of time in seconds, where x is in angstroms. What is the
average velocity of the particle from t = 2 to t = 8?
t
0
2
4
6
8
x(t)
0
72
92
144
180
3
Instantaneous Velocity
Definition 6 The instantaneous velocity is the slope of the distance function
at a point.
Average Velocity= s(a+h)−s(a)
h
Instantaneous Velocity (at t = a) = limh→0
s(a+h)−s(a)
h
Problem 3 Find the average velocity over the interval 0 ≤ t ≤ .2, and
estimate the velocity at t = 0.2 of a car whose position, s, is given by the
following table:
t (sec)
0
.2
.4
.6
.8
1.0
s (feet)
0
.5
1.8
3.8
6.5
9.6
Using Limits to Compute the Instantaneous Velocity
Example 5 Calculate the instantaneous velocity for s(t) = t2 at t = 3.
Solution.
4
METHOD 1. Estimate the limit numerically:
h
-.1
-.01
-.001
.001
.01
.1
(3+h)2 −9
h
5.9
5.99
5.999
6.001
6.01
6.1
METHOD 2. Use algebra to find the limit:
s(a + h) − s(a)
h→0
h
lim
(a + h)2 − a2
h→0
h
2
a + 2ah + h2 − a2
= lim
h→0
h
2ah + h2
= lim
h→0
h
h (2a + h)
= lim
h→0
h
= lim 2a + h = 2a
= lim
h→0
Now we use the fact that a = 3 to find that the instantaneous velocity
is 6.
H
Problem 4 In a time of t seconds, a particle moves a distance of s meters
from its starting point, where s = 3t2 .
(a) Find the average velocity between t = 1 and t = 1 + h if:
(i) h = 0.1
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(ii) h = 0.01
(iii) h = 0.001
(b) Use your answers to part (a) to estimate the instantaneous velocity of
the particle at time t = 1.
Problem 5 Use algebra to evaluate limh→0
Problem 6 Estimate limh→0
ues of h.
cos(h)−1
h
(1+h)3 −1
.
h
by substituting smaller and smaller val-
Problem 7 The graph of f (t) in the figure gives the position of a particle
at time t.
Estimate the following quantities:
A. average velocity between t = 1 and t = 3
B. average velocity between t = 5 and t = 6
C. instantaneous velocity at t = 1
D. instantaneous velocity at t = 3
6
E. instantaneous velocity at t = 5
F. instantaneous velocity at t = 6
Problem 8 A ball is tossed into the air from a bridge, and its height, y (in
feet), above the ground t seconds after it is thrown is given by the equation
y = f (t) = −16t2 + 50t + 36.
(a) How high above the ground is the bridge?
(b) What is the average velocity of the ball for the first second?
(c) Approximate the velocity of the ball at t = 1 second.
(d) Graph f , and determine the maximum height the ball reaches. What
is the velocity at the time the ball is at the peak?
(e) Use the graph to decide at what time, t, the ball reaches its maximum
height.
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