# Glencoe, Ch 2 Math Toolkit

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Math for Physics

Physics is an experimental science. In taking data during experiments, we need to be
aware of how “good” our data is, that is, the quality of our measuring devices.

When we design and conduct experiments, we are trying to determine the effect of one
variable on another. The independent variable is the one we have control over—we set
its values. The dependent variable is the response to the levels of the IV.

Once we have collected our data, we need to analyze it. We collect it in data tables,
then we often graph it to determine the relationship, if any, between our DV and our IV.

Quality of Data
We should be concerned about the quality of the data we take during experiments. Two
main questions:
1. How close are my measurements to the actual value? Accuracy
2. How close are repeat measurements to each other? Precision
3. Error analysis: ±5% is acceptable experimental error

Must explain error over ±5%. Not acceptable explanations: human error,
poor equipment. You need to think about causes of error and EXPLAIN their
effect on your results (eg: high humidity may have resulted in increased
measured mass, resulting in higher calculated force). If your error is very high, you
should repeat the experiment unless I have specifically told you that your result is okay.

Making a Graph
Manual: Set scale of axes to fill graph paper. Plot points as x. Look at overall shape of
data. Is it linear? If so, draw a “line of best fit” through it.
Line of best fit: line drawn so that distance between data points and line above line is
approx. equal to distance between data points and line below line. This line models
Excel: chart should almost always be an XY Scatterplot. Line of best fit is called a
trendline. Add it to your graph by clicking on Chart Layout  Trendline  Linear fit.
Then click on Chart Layout  More Options  Show equation (check box) and Show
R2 (check box). R2 is a measure of how close your data is to the line of best fit. R2 = 1
means every data point falls on the line of best fit (unlikely with experimental data).

Types of Relationships between IVs and DVs
No relationship—horizontal line or total scatter
Direct relationship—straight line of slope m (y=mx+b). The slope often has particular
meaning. Ex: slope of mass vs volume is density, slope of distance traveled vs
time is velocity
Indirect relationship—hyperbola (y=a/x). As x gets larger, y gets smaller. Graph would
be y vs 1/x so that a straight line is obtained.
Parabolic relationship—parabola (y=ax2). Projectile motion follows a parabolic curve.
Graph would be y vs x2.

Unit Conversions
Factor Label Method of Unit Conversion

Make a conversion factor from a known conversion:

1 kg = 1000 g          Divide both sides by 1000 g

1 kg   1000 g     1 kg
        1        1
1000 g 1000 g     1000 g

1 kg = 1000 g          Divide both sides by 1 kg

1 kg     1000 g         1000 g
1                       1
1 kg      1 kg           1 kg

When making a unit conversion, multiply your starting number (and unit) by the
appropriate conversion factor. The units undergo the same operations as the numbers.
If the units don’t make sense at the end, check that the conversion factor was not
inverted.

      Work Prob. 4-5 p. 21

Arithmetic Operations in Scientific Notation (Handout/Worksheet)

Review Laws of Exponents in Appendix A, p 741
Exponent of zero : a 0  1
Exponent of one : a1  a
1
Negative exp onents : a  n 
an

Operations with Exponents

Pr oduct of Powers : a m  a n  a m  n
Power of Powers : (a m ) n  a mn
am
Quotient of Powers :      n
 a mn
a
m
n  Root of Powers : n a m  a        n

Power of a Pr oduct : (ab) m  a mb m
Power of a Monomial : (a mb n ) p  a mp b np
Example: Simplify (2a4b)3[(-2b)3]2

Addition and Subtraction in Scientific Notation [M x 10n]
Three cases:
1. Numbers have same exponent: add or subtract M, keep same n.
2. Exponents are not the same: move decimal to left or right until they are the
same, then add or subtract M.
3. If one number has large exponent and the other has a small exponent, the
smaller number will not affect the value of the large number.

Multiplication and Division in Scientific Notation [M x 10n]
Perform operation on M, use laws of exponents to determine n.

Basic Trig
SOHCAHTOA

c
a
θ
b

sin θ = a / c              To find the angle θ when you know two sides of the triangle,
use the inverse functions:
cos θ = b / c
θ = sin-1 (a / c) = cos-1 (b / c) = tan-1 (a / b)
tan θ = a / b

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