Physics 101 Math Review. Solutions


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1 Physics 0 Math eview Slutins
2 . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t write than the rdinary number des. s an example 00 is easier t write than.00x0, but.00x0 8 is easier t write than 00,000,000). D yur best t cancel units, and attempt t shw the simplified units in the final answer. a. T s kg. 0 0 kg s 3.98x0  s b. K kg 0 4. (. m s).47x0 kg m /s c. F N m m 8.64x07 N d p P 3.04x0 e. e J J J 8.06x0  f. 33. sin sin g Kmax J s s.7 0 J h m s m s. Often prblems n the exam are dne with variables nly. Slve fr the variable indicated. Dn t let the different letters cnfuse yu. Manipulate them algebraically as thugh they were numbers. a. v v a s s, a v ( )  v 0 I g., r μ 0 I / (ss 0 ) r b. K kx, x (K/k) (/) m L mλl/x h. xm, d m d c. T p l g, g 4π l d. F G m m g, r Gm m r e. f. mgh mv, v T p x x vt at, t F g gh v v 0  (xx 0 )a a i. pv nt T n j. sin c, n k. l..5x09 J, pv/n c sin  (n /n ) qv mv, v f s s i, si qv/m f s 0
3 3. Science uses the KMS system (SI: System Internatinale). KMS stands fr kilgram, meter, secnd. These are the units f chice f physics. The equatins in physics depend n unit agreement. S yu must cnvert t KMS in mst prblems t arrive at the crrect answer. kilmeters (km) t meters (m) and meters t kilmeters gram (g) t kilgram (kg) centimeters (cm) t meters (m) and meters t centimeters millimeters (mm) t meters (m) and meters t millimeters elsius ( ) t Kelvin (K) atmspheres (atm) t Pascals (Pa) nanmeters (nm) t meters (m) and meters t nanmeters liters (L) t cubic meters (m 3 ) micrmeters ( m) t meters (m) Other cnversins will be taught as they becme necessary. What if yu dn t knw the cnversin factrs? lleges want students wh can d their wn infrmatin (s d emplyers). a g = kg h. 5.0 m.5x0 = m b.. km = 00 m i..65 mm =.65x03 m c. 83 nm = 8.3x07 m j. 8.3 m = 8.3x03 km d. 98 K = 5 k. 5.4 L = 5.4x03 m 3 e m = cm 77 l cm = 0.4 m f. 8.8x08 m = 8.8x05 mm m. 6.3x07 m = 63 nm g.. atm =.x0 5 Pa n..5x0 m =.5x0 8 km 6. Slve the fllwing gemetric prblems. a. Line tuches the circle at a single pint. Line extends thrugh the center f the circle. i. What is line in reference t the circle? Tangent ii. Hw large is the angle between lines and? 90 b. What is angle? c. What is angle? d. Hw large is?
4 e. The radius f a circle is 5.5 cm, i. What is the circumference in meters?.055**pi = 0.346m ii. What is its area in square meters?.055 *PI = 9.5x03 f. What is the area under the curve at the right? 4*+.5*4*8 = Using the generic triangle t the right, ight Triangle Trignmetry and Pythagrean Therem slve the fllwing. Yur calculatr must be in degree mde. 0 a. = 55 and c = 3 m, slve fr a and b. a= 8.4m, b= 6.m b. = 45 and a = 5 m/s, slve fr b and c. a= 0.6m/s, b= 0.6m/s c. b = 7.8 m and = 65, slve fr a and c. a= 8.30m, c= 9.6m Vectrs d. a = 50 m and b = 80 m, slve fr and c. 0 = 35.8, c = 308m e. a =5 cm and c = 3 cm, slve fr b and. b= 0cm, 0= 38.7 f. b =04 cm and c = 65 cm, slve fr a and. N slutin valid Mst f the quantities in physics are vectrs. This makes prficiency in vectrs extremely imprtant. Magnitude: Size r extent. The numerical value. Directin: lignment r rientatin f any psitin with respect t any ther psitin. Scalars: physical quantity described by a single number and units. quantity described by magnitude nly. Examples: time, mass, and temperature Vectr: physical quantity with bth a magnitude and a directin. directinal quantity. Examples: velcity, acceleratin, frce Ntatin: r Length f the arrw is prprtinal t the vectrs magnitude. Negative Vectrs Directin the arrw pints is the directin f the vectr. Negative vectrs have the same magnitude as their psitive cunterpart. They are just pinting in the ppsite directin. Vectr dditin and subtractin Think f it as vectr additin nly. The result f adding vectrs is called the resultant. + = S if has a magnitude f 3 and has a magnitude f, then has a magnitude f 3+=5. When yu need t subtract ne vectr frm anther think f the ne being subtracted as being a negative vectr. Then add them. 4
5 is really e j + e j = negative vectr has the same length as its psitive cunterpart, but its directin is reversed. S if has a magnitude f 3 and has a magnitude f, then has a magnitude f 3+()=. This is very imprtant. In physics a negative number des nt always mean a smaller number. Mathematically is smaller than +, but in physics these numbers have the same magnitude (size), they just pint in different directins (80 apart). There are tw methds f adding vectrs Parallelgram + Tip t Tail It is readily apparent that bth methds arrive at the exact same slutin since either methd is essentially a parallelgram. It is useful t understand bth systems. In sme prblems ne methd is advantageus, while in ther prblems the alternative methd is superir. 8. Draw the resultant vectr using the parallelgram methd f vectr additin. Example b. d. a. c. e. 9. Draw the resultant vectr using the tip t tail methd f vectr additin. Label the resultant as vectr Example : + a. X + Y Y X  Example : b. T S T S 5
6 c. P + V e. + + P V d. D f. D Directin: What des psitive r negative directin mean? Hw is it referenced? The answer is the crdinate axis system. In physics a crdinate axis system is used t give a prblem a frame f reference. Psitive directin is a vectr mving in the psitive x r psitive y directin, while a negative vectr mves in the negative x r negative y directin (This als applies t the z directin, which will be used sparingly in this curse). +y x +x y What abut vectrs that dn t fall n the axis? Yu must specify their directin using degrees measured frm East. mpnent Vectrs resultant vectr is a vectr resulting frm the sum f tw r mre ther vectrs. Mathematically the resultant has the same magnitude and directin as the ttal f the vectrs that cmpse the resultant. uld a vectr be described by tw r mre ther vectrs? Wuld they have the same ttal result? This is the reverse f finding the resultant. Yu are given the resultant and must find the cmpnent vectrs n the crdinate axis that describe the resultant. + y + y + x r + x ny vectr can be described by an x axis vectr and a y axis vectr which summed tgether mean the exact same thing. The advantage is yu can then use plus and minus signs fr directin instead f the angle. 0. Fr the fllwing vectrs draw the cmpnent vectrs alng the x and y axis. a. b. 6
7 c. d. Obviusly the quadrant that a vectr is in determines the sign f the x and y cmpnent vectrs. Trignmetry and Vectrs Given a vectr, yu can nw draw the x and y cmpnent vectrs. The sum f vectrs x and y describe the vectr exactly. gain, any math dne with the cmpnent vectrs will be as valid as with the riginal vectr. The advantage is that math n the x and/r y axis is greatly simplified since directin can be specified with plus and minus signs instead f degrees. ut, hw d yu mathematically find the length f the cmpnent vectrs? Use trignmetry x y cs adj hyp sin pp hyp adj hypcs pp hyp sin x hyp cs y hyp sin x 0cs40 y 0sin 40 x 766. y Slve the fllwing prblems. Yu will be cnverting frm a plar vectr, where directin is specified in degrees measured cunterclckwise frm east, t cmpnent vectrs alng the x and y axis. emember the plus and minus signs n yu answers. They crrespnd with the quadrant the riginal vectr is in. Hint: Draw the vectr first t help yu see the quadrant. nticipate the sign n the x and y vectrs. D nt bther t change the angle t less than 90. Using the number given will result in the crrect + and signs. The first number will be the magnitude (length f the vectr) and the secnd the degrees frm east. Yur calculatr must be in degree mde. Example: 50 at 35 b at 345 x hypcs 35 x 50cs35 x 43 x= 6.8 y= a. 89 at 50 y hyp sin y 50sin 35 y 05 c at 60 x= y=
8 d. 7.5x0 4 at 80 x= 7.5x0 4 y= 0 f. 990 at 30 x= 758 y= 636 e. at 65 x= .05 y= .95 g at 5 x= 69 y= 69. Given tw cmpnent vectrs slve fr the resultant vectr. This is the ppsite f number abve. Use Pythagrean Therem t find the hyptenuse, then use inverse (arc) tangent t slve fr the angle. Example: x = 0, y = 5 x y tan pp adj x y tan pp adj tan y x a. x = 600, y = 400 r= 7 0= 33.7 c. x = 3, y = 6 r= =6.6 b. x = 0.75, y = .5 r=.46 0= 59 d. x = , y = r= 0.0 0= 54. 8
9 e. x = 0,000, y = 4,000 f. x = 35, y = 998 Hw are vectrs used in Physics? They are used everywhere! Speed Speed is a scalar. It nly has magnitude (numerical value). v s = 0 m/s means that an bject is ging 0 meters every secnd. ut, we d nt knw where it is ging. Velcity ate Velcity is a vectr. It is cmpsed f bth magnitude and directin. Speed is a part (numerical value) f velcity. v = 0 m/s nrth, r v = 0 m/s in the +x directin, etc. There are three types f speed and three types f velcity Instantaneus speed / velcity: The speed r velcity at an instant in time. Yu lk dwn at yur speedmeter and it says 0 m/s. Yu are traveling at 0 m/s at that instant. Yur speed r velcity culd be changing, but at that mment it is 0 m/s. verage speed / velcity: If yu take a trip yu might g slw part f the way and fast at ther times. If yu take the ttal distance traveled divided by the time traveled yu get the average speed ver the whle trip. If yu lked at yur speedmeter frm time t time yu wuld have recrded a variety f instantaneus speeds. Yu culd g 0 m/s in a gas statin, r at a light. Yu culd g 30 m/s n the highway, and nly g 0 m/s n surface streets. ut, while there are many instantaneus speeds there is nly ne average speed fr the whle trip. nstant speed / velcity: If yu have cruise cntrl yu might travel the whle time at ne cnstant speed. If this is the case then yu average speed will equal this cnstant speed. trick questin Will an bject traveling at a cnstant speed f 0 m/s als always have cnstant velcity? Nt always. If the bject is turning arund a curve r mving in a circle it can have a cnstant speed f 0 m/s, but since it is turning, its directin is changing. nd if directin is changing then velcity must change, since velcity is made up f speed and directin. nstant velcity must have bth cnstant magnitude and cnstant directin. Speed and velcity are rates. rate is a way t quantify anything that takes place during a time interval. ates are easily recgnized. They always have time in the denminatr. 0 m/s 0 meters / secnd The very first Physics Equatin Velcity and Speed bth share the same equatin. emember speed is the numerical (magnitude) part f velcity. Velcity nly differs frm speed in that it specifies a directin. x v v stands fr velcity x stands fr displacement t stands fr time t Displacement is a vectr fr distance traveled in a straight line. It ges with velcity. Distance is a scalar and ges with speed. Displacement is measured frm the rigin. It is a value f hw far away frm the rigin yu are at the end f the prblem. The directin f a displacement is the shrtest straight line frm the lcatin at the beginning f the prblem t the lcatin at the end f the prblem. 9
10 Hw d distance and displacement differ? Suppses yu walk 0 meters dwn the + x axis and turn arund and walk 0 meters dwn the x axis. The distance traveled des nt depend n directin since it is a scalar, s yu walked = 30 meter. Displacement nly cares abut yur distance frm the rigin at the end f the prblem = 0 meter. 3. ttempt t slve the fllwing prblems. Take heed f the fllwing. lways use the MKS system: Units must be in meters, kilgrams, secnds. On the all tests, including the P exam yu must:. List the riginal equatin used.. Shw crrect substitutin. 3. rrive at the crrect answer with crrect units. Distance and displacement are measured in meters Speed and velcity are measured in meters per secnd Time is measured in secnds (m) (m/s) Example: car travels 000 meters in 0 secnds. What is its velcity? x 000m v v v 00m s t 0s a. car travels 35 km west and 75 km east. What distance did it travel? 0km (s) b. car travels 35 km west and 75 km east. What is its displacement? 8.8 km c. car travels 35 km west, 90 km nrth. What distance did it travel? 5km d. car travels 35 km west, 90 km nrth. What is its displacement? 96.6km e. bicyclist pedals at 0 m/s in 0 s. What distance was traveled? 00m f. n airplane flies 50.0 km at 300 m/s. Hw lng des this take? 833s g. skydiver falls 3 km in 5 s. Hw fast are they ging? 00m/s h. car travels 35 km west, 90 km nrth in tw hurs. What is its average speed? 6.5km/hr i. car travels 35 km west, 90 km nrth in tw hurs. What is its average velcity? 48.3km/hr 0
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