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2 x t2 2(205 m) (500 s)2 164 m or 164 m/s2 s2
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The number two is an exact number, so it does not affect the determination of significant digits Calculate and round to three significant digits
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Math Handbook
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Unit conversion Use a conversion factor to convert from one measurement unit to another of the same type, such as from minutes to seconds This is equivalent to multiplying by one Connecting Math to Physics Find x when v0 = 67 m/s and t = 50 min Use the equation, x v0 t
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67 m 50 min s 1
0 )( 16misn )
Multiply by the conversion factor
60 s 1 min
20100 m
20 104 m
Calculate and round to two significant digits The numbers 60 s and 1 min are exact numbers, so they do not affect the determination of significant digits
16 Simplify t
v a t and a
40 102 m 16 m/s
17 Find the velocity of a dropped brick after 50 s using
980 m/s2
0 18 Calculate the product: 32 cm 16misn 601 min 1010m cm 1s h
19 An Olympic record for 1000 m is 987 s What is the speed
in kilometers per hour
Dimensional Analysis
Dimensional analysis is a method of doing algebra with the units It often is used to check the validity of the units of a final result and the equation being used, without completely redoing the calculation Physics Example Verify that the final answer of df di is measured in m t is measured in s vi is measured in m/s a is measured in m/s2 df m m m m di vit
1 2 at will have the units m 2
( m )(s) s s (m)( ) s
(m)(1) m
1 1 m 2
1 m (s)2 2 s2 1 s2 (m) 2 2 s 1 (m)(1) 2
( ) ( )
Substitute the units for each variable Simplify the fractions using the distributive property Substitute s/s = 1, s2/s2 = 1 Everything simplifies to m, thus df is in m
The factor of in the above does not apply to the units It applies only to any number 2 values that would be inserted for the variables in the equation It is easiest to remove 1 number factors like the when first setting up the dimensional analysis
Math Handbook
VII Graphs of Relations
Math Handbook
The Coordinate Plane
Points are located in reference to two perpendicular number lines, called axes The horizontal number line is called the x-axis; the vertical number line is called the y-axis The x-axis represents the independent variable The y-axis represents the dependent variable A point is represented by two coordinates ( x, y), which also is called an ordered pair The value of the independent variable, x, always is listed first in the ordered pair The ordered pair (0, 0) represents the origin, which is the point where the two axes intersect
The vertical axis also is called the y-axis
(4, 3)
The coordinate system also is called the coordinate plane
Each point is named by an ordered pair 4 2 2 The origin, at (0, 0), is the point where the axes intersect 4 0 2 4
The horizontal axis also is called the x-axis
Graphing Data to Determine a Relationship
Use the following steps to graph data 1 Draw two perpendicular axes 2 Identify the independent and the dependent variables Label each axis using the variable names 3 Determine the range of data for each variable Use the ranges to decide on a convenient scale for each axis Mark and number the scales 4 Plot each data point 5 When the points seem to lie approximately in a line, draw a best-fit line through the points When the points do not lie in a line, draw a smooth curve through as many data points as possible When there does not appear to be a trend or a pattern to the dots, do US Dollars and British Pounds not draw any line or curve 70 6 Write a title that clearly describes what the 60 graph represents
Dollars 50 40 30 20 10 0 10 20 30 40 50 60 70 Pounds
Expense Hotel Meals Entertainment Transportation
Pounds 40 30 15 6
Dollars 58 43 22 9
Appendix A
Math Handbook
Interpolating and Extrapolating
Math Handbook
Interpolation is a process used to estimate a value for a relation that lies between two known values Extrapolation is a process used to estimate a value for a relation that lies beyond the known values The equation of a line through the points (the known values) helps you interpolate and extrapolate Example: Using the data and the graph, estimate the value, in dollars, of 20 pounds Use interpolation Locate two points on either side of 20 pounds 15 pounds and 30 pounds Draw a line (or place a straight edge) through the graphed points Draw a line segment from 20 on the x-axis to that line Draw a line segment from that intersection point to the y-axis Read the scale on the y-axis 20 pounds is about 28 or 29 dollars Example: Use extrapolation to estimate the value of 50 pounds Draw a line segment from 50 on the x-axis to the line through the points Extend the line if necessary Read the corresponding value on the y-axis Extend the axis scale if necessary 50 pounds is a little more than 70 dollars
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