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KINEMATICS LESSON 1 PART 1
Conversions
When doing physics problems sometimes you have to covert units such as miles to kilometers and seconds to hours.
Example:
1. Covert 56 000 seconds ( hours.
2. miles ( km
3. km ( inches
4. km/s ( mi/h
5. km2 ( mi2
6. m/s2 -> km/h2
and some more
Find HYPERLINK "http://oakroadsystems.com/math/" \l "Factors" the conversion factor for the given and desired units, and write it as a fraction with the given units in the opposite position from the original measurement. (If the original measurement has the given units in the numerator, the conversion fraction needs them in the denominator, and vice versa.) The value of that fraction is 1, since the top and bottom are equal.
If the given units are raised to a power, raise the conversion fraction to that same power.
Multiply the original measurement by the conversion fraction, and simplify.
Significant Numbers
What strikes you about the difference between a measurement of 1.615in and 1.6in?
The first is more precise than the second, meaning it could be anything between 1.6145 and 1.6154.
ROUNDING
Example: Round the following:
3.4674
2.45
3.23
89.906
The significant digits in a number start at the first non-zero digit and end at the last digit.
Examples: 1417 has four significant digits, and so do 1.417 and 0.00001417.
What about 14.1700? It has six significant digits, not four, because only zeroes at the start of a number are non-significant.
Finally, 14.07 has four significant digits.
There can be some ambiguity with trailing zeroes in a large whole number. For instance, we quote the average distance from earth to sun as 93 million miles. In that form, the number has two significant digits. All were saying is that the average distance is 92 to 93 million.
To get around this problem, numbers are often expressed in scientific notation. For instance, the figure of 93 million miles is 9.3107 miles (nine point three times ten to the seventh). On your calculator it appears as 9.3E7.
Practice
How many significant digits are in each of these numbers? 4800, 4800.0, 4.8, 0.0000067, .0000067.
Answer: Note 4800, zeros are probably just place holders, but 4800.0 the last zero is not needed to hold a place, therefore significant.
Rounding the results of Calculations
Now you see whats wrong with 25 feet divided by 6.0 equals 4.166666667 feet, which is what your calculator will tell you.
A measurement of 25 feet is accurate only to the nearest foot; you cant get an answer that is accurate to a billionth of a foot!
Significant Digits in Multiplication, Division, Trig. functions, etc.
In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.
Example:
1. Compute and round your answer properly: 34.7811.70.17. Answer: the least significant number has two significant digits, and therefore the answer must have two significant digits. You take the 2393.682353 from your calculator and round it to 2400 (or 2.4102 in scientific notation).
2. Compute and round your answer properly: 16.2. Answer: the original has three significant digits, and the answer must have the same. Your calculator gives 262.44, which you round to 262.
3. Three people will share a lottery prize of $24.8 million equally. How much will each one receive, before taxes? Answer: 24.83= 8.2666666667 on the calculator. 24.8 has three significant digits. The 3 looks like it has only one significant digit, but it is an exact number: not between 2 and 3 people but 3 people exactly. Think of it as being 3.000000000... people. Therefore the answer will have three significant digits: each person gets $8.27 million.
Significant Digits in Addition and Subtraction
When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted.
Example:
5.67 J (two decimal places) 1.1 J (one decimal place) 0.9378 J (four decimal place) 7.7 J (one decimal place)
Keep One Extra Digit in Intermediate Answers
When doing multi-step calculations, keep at least two more significant digit in intermediate results than needed in your final answer.
For instance, if a final answer requires two significant digits, then carry at least four significant digits in calculations. If you round-off all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect.
You are usually safe if you compute with two more significant figures than your ultimate answer will have.
Discuss Scientific Notation - give examples
Significant Figures and Conversion HOMEWORK
Conversion Chart
1 centimeter = 0.3937 inches
12 inches = 1 foot
1 inch = 2.54 centimeters
1 foot = 0.3048 meters
1 kilometer = 0.6214 miles
1 kg = 2.024 lbs
ROUNDING
1. Round to the nearest whole number: 17.514, 24.500, 24.501, 27.499.
2. Round to one decimal place: 17.545, 24.451, 38.989.
3. How many significant numbers are in the following numbers:
0.000 000 34
450 000
23.00
65.7002
22.980
.0980
3 puppies
00079.32
4. Answer the following questions using the proper significant figures in the answers.
81.72.405 = ?
ekt = ?, where k = 0.0189 yr-1, and t = 25 yr.
ab/c = ?, where a = 483 J, b = 73.67 J, and c = 15.67
x + y + z = ?, where x = 48.1, y = 77, and z = 65.789
m - n - p = ?, where m = 25.6, n = 21.1, and p = 2.43
5. A goat is tethered to a post in your back yard with a chain 8.4 feet long. Assuming the goat doesnt eat the chain, how much area of your yard will the goat be able to munch on? (Recall t h a t t h e a r e a o f a c i r c l e i s r . )
C O N V E R S I O N S ( s h o u l d h a v e s i g f i g s )
R o u n d a n s w e r s t o s i g n i f i c a n t f i g u r e s .
6 . U s e t h e c h a r t a n d a c a l c u l a t o r t o c o n v e r t e a c h m e a s u r e m e n t . B e s u r e t o s h o w y o u r w o r k .
a . 1 6 i n = _ _ _ _ _ _ _ c m
b . 3 4 5 l b s = _ _ _ _ _ _ _ k g
c . 4 5 0 km = _______ mi
d. 40 m = _______ ft
e. 50 kg = _______ lbs
7. Penny has a pencil that is 19 cm long. How long the pencil in inches?
8. Macho Mel can lift 200 kilograms with ease. How much is this in pounds?
9. The distance between Happyville and Giggleland is 60 miles. How far is this in kilometers?
10. Escape velocity from the earths surface is about 7.0 mi/sec. What is that in mi/hr?
11. A race car has a top speed of 310 km/hr. What is that in m/sec?
12. Complete the following table showing conversion steps:
Object Speed (m/s) Speed (km/h)
a. Marathon Runner 4.2 m/s ( ) = __________
b. Ashalt Paving Machine ___________ = 0.80 km/h ( )
c. Indy Car 90.0 m/s ( ) = __________
d. Mach 1.5 Jet ___________ = 1800 km/h ( )
e. School Zone 8.3 m/s ( ) = __________
f. Lake Canoeing ___________ = 2.0 km/h ( )
13. If my apartment measures 850 square feet, what is that in square meters? In other words, convert 850ft to m.
14. How many cubic meters are there in a cubic mile? (1 mi = 1609.344m.)
15. Along the way to school, Anna Lytical calculated that she walked 4 miles North and 2 km East to get to School. How far did she walk in total?
16. My 1967 Encyclopdia Britannica says that Lake Erie has a surface area of 9930 square miles and an average depth of 58 feet. How much water does it hold, in cubic miles? i n l i t e r s ? ( 1 m i = 5 2 8 0 f t ; 1 l i t e r = 0 . 0 0 1 m , a n d u s e t h e a n s w e r t o t h e p r e v i o u s p r o b l e m . )
A N S W E R K E Y
1 7 . 5 1 4 ! 1 8 ( 5 t o 9 , r o u n d u p ) , 2 4 . 5 0 0 ! 2 5 ( 5 t o 9 , r o u n d u p ) , 2 4 . 5 0 1 ! 2 5 ( 5 t o 9 , r o u n d u p ) , 2 7 . 4 9 9 ! 2 7 ( 0 t o 4 , r o u n d d o w n ) .
1 7 . 5 4 5 ! 1 7 . 5 ( 0 t o 4 , r o u n d d o w n ) , 2 4 . 4 5 1 ! 2 4 . 5 ( 5 t o 9 , r o u n d u p ) , 3 8 . 9 8 7 ! 3 9 . 0 ( 3 8 . 9 | 8 7 t h e 8 o f 8 7 m e a n s t h a t t h e 9 o f 3 8 . 9 m u s t i n c r e a s e ; 3 8 . 9 r o l l s o v e r t o 3 9 . 0 ) .
a . 2 b . 2 c . 4 d . 6 e . 5 f . 3 g . i n f i n i t y h . 4
a . 1 9 6 b . 1 . 6 c . 2 . 2 7 X 1 0 3 J 2 d . 191 e. 2.1
220 ft2
a. 6.3 b. 156 c. 280 d. 1 x 102 e. 1 x 102
7.4 in
4 x 102 lbs
1 x 102 km
25,000 mi/hr
86 m/s
a. 15 km/h b. 0.22 m/s c. 324 km/h d. 5.0 x 102 m/s e. 3.0 x 101 km/h f. 0.56 m/s
79 m2
1 mi = 4.168182109 (about 4168 million) m.
8 km or 5 mi
109 mi, 4.551014 liters.
HOMEWORK
Conversion Chart
1 centimeter = 0.3937 inches
12 inches = 1 foot
1 inch = 2.54 centimeters
1 foot = 0.3048 meters
1 kilometer = 0.6214 miles
1 kg = 2.024 lbs
ROUNDING
1. Round to the nea r e s t w h o l e n u m b e r : 1 7 . 5 1 4 , 2 4 . 5 0 0 , 2 4 . 5 0 1 , 2 7 . 4 9 9 .
A n s w e r s : 1 7 . 5 1 4 ! 1 8 ( 5 t o 9 , r o u n d u p ) , 2 4 . 5 0 0 ! 2 5 ( 5 t o 9 , r o u n d u p ) , 2 4 . 5 0 1 ! 2 5 ( 5 t o 9 , r o u n d u p ) , 2 7 . 4 9 9 ! 2 7 ( 0 t o 4 , r o u n d d o w n ) .
2 . R o u n d t o o n e d e c i m a l p l a c e : 1 7 . 5 4 5 , 2 4 . 4 5 1 , 3 8 . 9 8 9 .
A n s w e r s : 1 7 . 5 4 5 ! 1 7 . 5 ( 0 t o 4 , r o u n d d o w n ) , 2 4 . 4 5 1 ! 2 4 . 5 ( 5 t o 9 , r o u n d u p ) , 3 8 . 9 8 7 ! 3 9 . 0 ( 3 8 . 9 | 8 7 t h e 8 o f 8 7 m e a n s t h a t t h e 9 o f 3 8 . 9 m u s t i n c r e a s e ; 3 8 . 9 r o l l s o v e r t o 3 9 . 0 ) .
3 . H o w m a n y s i g n i f i c a n t n u m b e r s a r e i n t h e f o l l o w i n g n u m b e r s :
0 . 0 0 0 0 0 0 3 4
4 5 0 0 0 0
23.00
65.7002
22.980
.0980
3 puppies
00079.32
a. 2 b. 2 c. 4 d. 6 e. 5 f. 3 g. infinity h. 4
4. Answer the following questions using the proper significant figures in the answers.
81.72.405 = ? Answer: Your calculator says 196.4805. But 81.7 has three significant digits and 2.405 has four; therefore the answer must have three significant digits (smaller of 3 and 4). 81.72.405= 196.
ekt = ?, where k = 0.0189 yr-1, and t = 25 yr. [Ans. 1.6]
ab/c = ?, where a = 483 J, b = 73.67 J, and c = 15.67 [Ans. 2.27X103 J2 ]
x + y + z = ?, where x = 48.1, y = 77, and z = 65.789 [Ans. 191]
m - n - p = ?, where m = 25.6, n = 21.1, and p = 2.43 [Ans. 2.1]
5. A goat is tethered to a post in your back yard with a chain 8.4 feet long. Assuming the goat doesnt eat the chain, how much area of your yard will the goat be able to munch on? (Recall that the area o f a c i r c l e i s r . )
A n s w e r : 8 . 4 h a s t w o s i g n i f i c a n t d i g i t s , a n d t h e r e f o r e t h e a n s w e r w i l l a l s o h a v e t w o s i g n i f i c a n t d i g i t s . Y o u n e e d 8 . 4 . E n t e r 8 . 4 , t h e n p r e s s t h e [ x ] k e y a n d [ E N T E R ] t o o b t a i n 7 0 . 5 6 . N o w p r e s s t h e [ ] k e y . T o e n t e r , f i n d t h e s y m b o l i n g o l d j u s t o v e r t h e [ ^ ] k e y ( w h i c h i s a b o v e t h e [ ] k e y ) . P r e s s [ 2 n d ] [ ] a n d [ E N T E R ] t o o b t a i n 2 2 1 . 6 7 0 7 7 7 6 ; r o u n d t o t w o s i g n i f i c a n t d i g i t s f o r a n a n s w e r o f 2 2 0 s q u a r e f e e t .
C O N V E R S I O N S ( s h o u l d h a v e s i g f i g s )
R o u n d a n s w e r s t o s i g n i f i c a n t f i g u r e s .
6 . Use the chart and a calculator to convert each measurement. Be sure to show your work.
a. 16 in = _______ cm
b. 345 lbs = _______ kg
c. 450 km = _______ mi
d. 40 m = _______ ft
e. 50 kg = _______ lbs
Answer: a. 6.3 b. 156 c. 280 d. 1 x 102 e. 1 x 102
7. Penny has a pencil that is 19 cm long. How long the pencil in inches?
Answer: 7.4 in
8. Macho Mel can lift 200 kilograms with ease. How much is this in pounds?
4 x 102 lbs
9. The distance between Happyville and Giggleland is 60 miles )
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hour
7.0 mi x 3600s = 25200 mi/h
s h
answer: 25,000 mi/hr (2 significant figures)
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1 hour = 3600 seconds
1 km = 1000 m
310 km x 1000 m x 1 h = 86 m/s
h 1 km 3600 s
12. Complete the following table showing conversion steps:
Object Speed (m/s) Speed (km/h)
a. Marathon Runner 4.2 m/s ( ) = __________
b. Ashalt Paving Machine ___________ = 0.80 km/h ( )
c. Indy Car 90.0 m/s ( ) = __________
d. Mach 1.5 Jet ___________ = 1800 km/h ( )
e. School Zone 8.3 m/s ( ) = __________
f. Lake Canoeing ___________ = 2.0 km/h ( )
Answer: a. 15 km/h b. 0.22 m/s c. 324 km/h d. 5.0 x 102 m/s e. 3.0 x 101 km/h f. 0.56 m/s
13. If my apartment measures 850 square feet, what is that in square meters? In other words, convert 850ft to m.
1 ft = 12 inches
1 m = 39.37 inches
1 m x 12 in = 12 m = 0.3048 m
39.37 in 1 ft 39.37 ft ft
850 ft2 x (0.3048 m)2
ft
850 ft2 x 0.30482 m2 = 79 m2
ft2
14. How many cubic meters are there in a cubic mile? (1 mi = 1609.344m.)
Answer. 1 mi = 4.168182109 (about 4168 million) m.
15. Along the way to school, Anna Lytical calculated that she walked 4 miles North and 2 km East to get to School. How far did she walk in total?
Answer: 8 km or 5 mi
16. My 1967 Encyclopdia Britannica says that Lake Erie has a surface area of 9930 square miles and an average depth of 58 feet. How much water does it hold, in cubic miles? in liters? (1 mi = 5280 ft; 1liter= 0.001m, and use the answer to the previous problem.)
Answer: 109 mi, 4.551014 liters.
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