Calculation of Velocity

Note that speed is an everyday term. It is never negative. In the above graph, the speed is given by the magnitude (absolute value) of the slope m. In contrast, velocity is a technical term. In can be positive, zero or negative. In a distance d versus time t graph, the velocity v is given by the slope m, and can be positive, zero or negative.

A graph of Barbara's position versus time follows. Find her velocity for each portion of her journey }

if the coordinates of the ends of each line segment in the above graph are as follows.

Solution. The rise over run ratio gives the following slope m_j on the j-th interval for 1 £ j £ 5, that is for j=1,2,3,4 or 5. The calculations ad nauseum follow.

 

m1 = d2-d1 t2-t1 = 40 km-80 km 30 min - 0min = - 40 30 km min = - 4 3 km min m2 = d3-d2 t3-t2 = 120 km-40 km 60 min -30min =     80 3 km min =     8 3 km min m3 = d4-d3 t4-t3 = 120 km-120 km 90 min -60min =     0 30 km min =     0 m4 = d5-d4 t5-t4 = 90 km-120 km 150min- 90min = - 30 60 km min = - 1   km min m5 = d6-d5 t6-t5 = 0 km-90 km 210min-150min = - 90 60 km min = - 3 2 km min

 

Content Guide
Foreword
Chapter Descriptions
1. Introduction
Cal. Preview (1983  lesson why slopes)
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units

Enriched material: The Appendices of Volume 3 are located in the Real  Analysis  Area.

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity

These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem is a postscript,
not in printed version.

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