This document contains examples and problems related to velocity and falling bodies. It discusses calculating average velocity over different time periods for an object falling from the CN Tower. It also contains examples of using the geometric approach to find the equation of a tangent line to a parabola at a given point by approximating the slope as the limit of a secant line.

1. The Tangent and 2.1 Velocity Problems

2. 2
Ticket
I got a ticket on the way home from DC. The trip was 230
miles long. It took me 4 hours.
What was my average speed?
Can I use this to argue the ticket? Why?

3. 3
Falling Bodies
Galileo:
s(t) = 4.9t 2
s is the distance fallen in meters
t is the time fallen in seconds

4. Example 3 – Velocity of a Falling Ball
Suppose that a ball is
dropped from the
upper observation
deck of the CN Tower
in Toronto, 450 m
above the ground.
Find the velocity of
the ball after 5
seconds.
4

5. 5
Velocity of Falling Ball
Find average velocity for first 6 seconds:

6. 6
Velocity of Falling Ball
Find average velocity from 5 to 6 seconds.

7. 7
Velocity of Falling Ball
Find average velocity from 5 to 5.1 seconds.

8. 8
Example 3 – Solution
The following table shows the results of similar calculations
of the average velocity over successively smaller time
periods.
cont’d

9. 9
Algebraic Approach

10. 10
Falling Bodies - Example
In English units: s(t) = 16t 2 s in feet, t in seconds
Use the Algrebraic approach to find velocity at 4 seconds

11. 11
Geometric Approach
Instantaneous rate of change
 slope of tangent line (just touches graph)
 Limiting case of secant line as 2 points get close

12. Example 1
Find an equation of the tangent line to the parabola y = x2 at
the point P(1, 1).
12
Solution:
We will be able to find an equation of the tangent line t as
soon as we know its slope m.
The difficulty is that we know only one point, P, on t,
whereas we need two points to compute the slope.

13. Example 1 – Solution
approximate m by choosing a point h units away on x-axis
Q(1+h, (1+h)2) on the parabola and compute the slope mPQ of
the secant line PQ.
cont’d
13

14. Example 1 – Solution cont’d
14

15. 15
Tangent Line Examples
(a) Find equation of tan to y = x3 at x = 1.
(b) Find equation of tan to y = 2 – x2 at x = 3