Name__________________________
Date__________________________
Topic 2.1-C Velocity in function notation
Our goal in this lesson is to follow up on the velocity in function notation lecture with
some experience.
x(t + t) - x(t)
v =
t
The above formula for average velocity is basically the result of the above-named
lecture. Although you may feel, deep down inside, that it is a less convenient formula
than v = x/t, you will see later why it really is a more convenient formula for finding
instantaneous velocity. For the following problems we will be using our usual sample
function for position vs. time x(t) = 2t2. Assume EXACT values.
1)
Complete the table below for finding the average velocity near t = 0 s:
t
t
0
0.001
2)
t
1
0.001
t
2
0.001
x(t + t) - x(t)
t
t + t
x(t)
x(t + t)
x(t + t) - x(t)
t
t + t
x(t)
x(t + t)
x(t + t) - x(t)
t
Complete the table below for finding the average velocity near t = 3 s:
t
t
3
0.001
5)
x(t + t)
Complete the table below for finding the average velocity near t = 2 s:
t
4)
x(t)
Complete the table below for finding the average velocity near t = 1 s:
t
3)
t + t
t + t
x(t)
x(t + t)
x(t + t) - x(t)
t
Complete the table below for finding the average velocity near t = 4 s:
t
t
4
0.001
t + t
x(t)
x(t + t)
x(t + t) - x(t)
t
6)
Why can we say that these average velocities are close to the instantaneous
velocities?
v (m/s)
7)
Complete the table for instantaneous velocity v vs. t, using your results from
problems 1) through 5). Then graph your results.
16
t
v
14
(s)
(m/s)
12
0
10
8
1
6
2
4
2
3
0
0
2
3
4
1
4
t (s)
8)
What is the slope? ________. What is the equation of the graph? v = ___t.
9)
Now let's do some letter-crunching instead of number-crunching. Do not
substitute values for t or t. Complete the table below:
substituted simplified
x(t+t)
x(t)
x(t+t) - x(t)
x(t+t) - x(t)
t
10)
Write your simplified formula for v here: v =
11) The above formula is a formula for the average velocity at any t, given any t.
Complete the table below using this formula:
t
t
0
0.001
1
2
3
4
0.001
0.001
0.001
0.001
v(substituted)
v(simplified)
12)
How do the results of 11) compare with the results of 1) through 5)? __________.
13)
Suppose t is zero in the formula from 10). Compare to the results from 8).