Physics Intro & Kinematics

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Physics Intro &Kinematics Quantities Velocity Units Acceleration Vectors Kinematics Displacement Graphing Motion in 1-DMass vs. WeightMass Scalar (no direction) Measures the amount of matter in an objectWeight Vector (points toward center of Earth) Force of gravity on an objectOn the moon, your mass would be the same,but the magnitude of your weight would be less.Some Physics QuantitiesVector - quantity with both magnitude (size) and directionScalar - quantity with magnitude onlyVectors: DisplacementScalars: Distance Velocity Acceleration Momentum Force Speed Time Mass EnergyVectorsVectors are represented with arrows The length of thearrow represents themagnitude (how far,how fast, how strong,etc, depending on thetype of vector).5 m/s42 UnitsSI PrefixesUnits are not the same as quantities!Quantity . . . Unit (symbol) Displacement & Distance . . . meter (m) Time . . . second (s) Velocity & Speed . . . (m/s) Acceleration . . . (m/s2) Mass . . . kilogram (kg) Momentum . . . (kg·m/s) Force . . .Newton (N) Energy . . . Joule (J) The arrow points inthe directions of theforce, motion,displacement, etc. Itis often specified byan angle.Little GuysBig Guysnanop 10-12n -2picokilok103megaM1061

Kinematics definitions Kinematics – branch of physics; studyof motion Position (x) – where you are located Distance (d ) – how far you havetraveled, regardless of direction Displacement ( x) – where you are inrelation to where you startedDistance vs. Displacement You drive the path, and your odometer goes upby 8 miles (your distance). Your displacement is the shorter directeddistance from start to stop (green arrow). What if you drove in a circle?startstopSpeed, Velocity, & Acceleration Speed (v) – how fast you go Velocity (v) – how fast and which way;the rate at which position changes Average speed ( v ) – distance / time Acceleration (a) – how fast you speedup, slow down, or change direction;the rate at which velocity changesSpeed vs. Velocity Speed is a scalar (how fast something ismoving regardless of its direction).Ex: v 20 mph Speed is the magnitude of velocity. Velocity is a combination of speed anddirection. Ex: v 20 mph at 15 south of west The symbol for speed is v. The symbol for velocity is type written in bold: vor hand written with an arrow: vSpeed vs. Velocity During your 8 mi. trip, which took 15 min., yourspeedometer displays your instantaneous speed,which varies throughout the trip. Your average speed is 32 mi/hr. Your average velocity is 32 mi/hr in a SEdirection. At any point in time, your velocity vector pointstangent to your path. The faster you go, the longer your velocity vector.AccelerationAcceleration – how fast you speed up, slowdown, or change direction; it’s the rate atwhich velocity changes. Two examples:t (s)v (mph)t (s)v (m/s)055034157131259228361325a 2 mph / sa -3 m/s -3 m/s 2s2

Acceleration due to GravityVelocity & Acceleration Sign ChartVELOCITYACCELERATION - -Moving forward;Moving backward;Speeding upSlowing downMoving forward;Moving backward;Slowing downSpeeding upNear the surface of theEarth, all objectsaccelerate at the samerate (ignoring airresistance).a -g -9.8 m/s2This accelerationvector is thesame on the wayup, at the top,and on the waydown!9.8 m/s2Interpretation: Velocity decreases by 9.8 m/s each second,meaning velocity is becoming less positive or morenegative. Less positive means slowing down while goingup. More negative means speeding up while going down.Kinematics DerivationsKinematics Formula SummaryFor 1-D motion with constant acceleration:a v/ t (by definition)a (vf – v0) / tvf v0 at vf v0 a t v (v0 vf )/ 2a vg1 x v0 t 2 a t 2v (v0 vf )/2 will be proven when we do graphing.av g½ vf2 – v02 2 a x(derivations to follow)Kinematics Derivations (cont.)vf v0 at x v0 t t (vf – v0)/a1 2at2 x v0 [(vf – v0)/a] 12a[(vf – v0)/a] 2vf2 – v02 2a xNote that the top equation is solved for t and thatexpression for t is substituted twice (in red) into the x equation. You should work out the algebra to provethe final result on the last line. x v t ½ (v0 vf) t ½ (v0 v0 a t) t1 x v0 t 2 a t 2(cont.)Sample Problems1. You’re riding a unicorn at 25 m/s and come toa uniform stop at a red light 20 m away.What’s your acceleration?2. A brick is dropped from 100 m up. Find itsimpact velocity and air time.3. An arrow is shot straight up from a pit 12 mbelow ground at 38 m/s.a. Find its max height above ground.b. At what times is it at ground level?3

Graphing !xMulti-step ProblemsB1. How fast should you throw a kumquatstraight down from 40 m up so that itsimpact speed would be the same as amango’s dropped from 60 m?1 – D MotionAtCAnswer: 19.8 m/s2. A dune buggy accelerates uniformly at1.5 m/s2 from rest to 22 m/s. Then thebrakes are applied and it stops 2.5 slater. Find the total distance traveled.A Starts at home (origin) and goes forward slowlyB Not moving (position remains constant as timeprogresses)C Turns around and goes in the other directionquickly, passing up homeAnswer: 188.83 mxBCGraphing w/AccelerationTangentLinesxttADOn a position vs. time graph:A Start from rest south of home; increase speed graduallySLOPEVELOCITYSLOPESPEEDB Pass home; gradually slow to a stop (still moving north)PositivePositiveSteepFastC Turn around; gradually speed back up again heading southNegativeNegativeGentleSlowZeroZeroFlatZeroD Continue heading south; gradually slow to a stop near thestarting pointIncreasing &DecreasingxxConcavityttIncreasingDecreasingOn a position vs. time graph:On a position vs. time graph:Increasing means moving forward (positive direction).Concave up means positive acceleration.Decreasing means moving backwards (negativedirection).Concave down means negative acceleration.4

xQRPSpecialPointsCurveSummaryxBCtStAInflection Pt.P, RChange of concavityPeak or ValleyQTurning pointTime AxisInterceptP, STimes when you are at“home”IncreasingDecreasingAll 3 Graphsxtavv 0a 0 (D)v 0a 0 (C)Car AnimationttConcave Downv 0a 0 (B)This website will allow you to set the initialvelocity and acceleration of a car. As the carmoves, all three graphs are generated.vGraphing TipsConcave Upv 0a 0 (A)Graphing Animation LinktxDGraphing TipsThe same rules apply in making an acceleration graph from avelocity graph. Just graph the slopes! Note: a positive constantslope in blue means a positive constant green segment. Thesteeper the blue slope, the farther the green segment is from thetime axis.vt Line up the graphs vertically. Draw vertical dashed lines at special points except intercepts. Map the slopes of the position graph onto the velocity graph.tat A red peak or valley means a blue time intercept.5

Area under a velocity graphReal lifeNote how the v graph is pointy and the a graph skips. In reallife, the blue points would be smooth curves and the greensegments would be connected. In our class, however, we’llmainly deal with constant acceleration.v“forward area”tv“backward area”tArea above the time axis forward (positive) displacement.aArea below the time axis backward (negative) displacement.tNet area (above - below) net displacement.Total area (above below) total distance traveled.v“forward area”v (m/s)Area12Area unitstt (s)“backward area”The areas above and below are about equal, so eventhough a significant distance may have been covered, thedisplacement is about zero, meaning the stopping point wasnear the starting point. The position graph shows this too.xtGraphs of a ballthrown straight upxThe ball is thrown fromthe ground, and it landson a ledge.tvThe position graph isparabolic.tThe ball peaks at theparabola’s vertex.The v graph has aslope of -9.8 m/s2.atMap out the slopes!There is more “positivearea” than negative onthe v graph. Imagine approximating the areaunder the curve with very thin12 m/srectangles. Each has area of height width.0.5 s The height is in m/s; width is inseconds. Therefore, area is in meters! The rectangles under the time axis have negativeheights, corresponding to negative displacement.Graph PracticeTry making all three graphs for the following scenario:1. Schmedrick starts out north of home. At time zero he’sdriving a cement mixer south very fast at a constant speed.2. He accidentally runs over an innocent moose crossingthe road, so he slows to a stop to check on the poor moose.3. He pauses for a while until he determines the moose issquashed flat and deader than a doornail.4. Fleeing the scene of the crime, Schmedrick takes offagain in the same direction, speeding up quickly.5. When his conscience gets the better of him, he slows,turns around, and returns to the crash site.6

Uniform AccelerationKinematics Practice x 1A catcher catches a 90 mph fast ball. Hisglove compresses 4.5 cm. How long does ittake to come to a complete stop? Be mindfulof your units!2.24 msAnswerSpreadsheet Problem Explain your answermathematically.t (s) x (m)0018.66234.64377.944138.56av0delta x2(m) ratio (m/s) (m/s )08.66125.98343.30560.62711 x 524 x 739( arbitrary units )416When object starts from rest and undergoes constantacceleration: Position is proportional to the square of time. Position changes result in the sequence of oddnumbers. Falling bodies exhibit this type of motion (since gis constant).Relationships We’re analyzing position as a function of time, initialvelocity, and constant acceleration. x, x, and the ratio depend on t, v0, and a. x is how much position changes each second. The ratio (1, 3, 5, 7) is the ratio of the x’s. Make a spreadsheetlike this and determinewhat must be trueabout v0 and/or a inorder to get this ratioof odd numbers.t:0x:0 x 317.3Let’s use the kinematics equations to answer these:1. A mango is dropped from a height h.a. If dropped from a height of 2 h, would theimpact speed double?b. Would the air time double when dropped froma height of 2 h ?2. A mango is thrown down at a speed v.a. If thrown down at 2 v from the same height,would the impact speed double?b. Would the air time double in this case?3. A rubber chicken is launched straightup at speed v from ground level.Find each of the following if thelaunch speed is tripled (in terms ofany constants and v).Re la ti ons hip s (con t.)a. max height9v2 / 2g6v / gb. hang timec. impact speed 3vAnswers7

Physics Intro & Kinematics Quantities Units Vectors Displacement Velocity Acceleration Kinematics Graphing Motion in 1-D Some Physics Quantities Vector - quantity with both magnitude (size) and direction Scalar - quantity with magnitude only Vectors: Displacement Velocity Acceleration Momentum Force Scalars: Distance Speed Time Mass .