A Mathematical Model of MotionCHAPTER 5

PHYSICS

5.1 Graphing Motion in One DimensionInterpret graphs of position versus time for a moving object to determine the velocity of the objectDescribe in words the information presented in graphs and draw graphs from descriptions of motionWrite equations that describe the position of an object moving at constant velocity

Parts of a GraphX-axisY-axisAll axes must be labeled with appropriate units, and values.

5.1 Position vs. TimeThe x-axis is always timeThe y-axis is always positionThe slope of the line indicates the velocity of the object.Slope = (y2-y1)/(x2-x1)d1-d0/t1-t0d/t

Uniform MotionUniform motion is defined as equal displacements occurring during successive equal time periods (sometimes called constant velocity)Straight lines on position-time graphs mean uniform motion.

Given below is a diagram of aball rolling along a table. Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds.Notice that the ball covers an equal distance between flashes. Let's assume this distance equals 20 cm and display the ball's behavior on a graph plotting its x-position versus time.

The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This represents the ball's average velocity as it moves across the table. Since the ball is moving in a positive direction its velocity is positive. That is, the ball's velocity is a vector quantity possessing both magnitude (200 cm/sec) and direction (positive).

Steepness of slope on Position-Time graphSlope is related to velocity Steep slope = higher velocityShallow slope = less velocity

Different Position. Vs. Time graphsConstant positive velocity(zero acceleration)Constant negative velocity(zero acceleration)Increasing positive velocity(positive acceleration)Decreasing negative velocity(positive acceleration)Uniform MotionAccelerated Motion

Different Position. Vs. TimeChanging slope means changing velocity!!!!!!Decreasing negative slope = ??Increasing negative slope = ??

A Starts at home (origin) and goes forward slowlyB Not moving (position remains constant as time progresses)C Turns around and goes in the other direction quickly, passing up home

During which intervals washe traveling in a positive direction?During which intervalswas he traveling in a negative direction?During which intervalwas he resting in a negative location?During which intervalwas he resting in a positive location?During which two intervals did he travel at the same speed?A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec

Graphing w/ AccelerationxA Start from rest south of home; increase speed graduallyB Pass home; gradually slow to a stop (still moving north)C Turn around; gradually speed back up again heading southD Continue heading south; gradually slow to a stop near the starting point

You try it..Using the Position-time graph given to you, write a one or two paragraph story that describes the motion illustrated.You need to describe the specific motion for each of the steps (a-f)You will be graded upon your ability to correctly describe the motion for each step.

Tangent Linest xOn a position vs. time graph:

SLOPEVELOCITYPositivePositiveNegativeNegativeZeroZero

SLOPESPEEDSteepFastGentleSlowFlatZero

Increasing & DecreasingIncreasingDecreasingOn a position vs. time graph:Increasing means moving forward (positive direction).Decreasing means moving backwards (negative direction).

ConcavityOn a position vs. time graph:Concave up means positive acceleration.Concave down means negative acceleration.

Special PointsPQRS

Inflection Pt. P, RChange of concavity, change of acceleration Peak or ValleyQTurning point, change of positive velocity to negativeTime Axis InterceptP, STimes when you are at home, or at origin

5.2 Graphing Velocity in One DimensionDetermine, from a graph of velocity versus time, the velocity of an object at a specific timeInterpret a v-t graph to find the time at which an object has a specific velocityCalculate the displacement of an object from the area under a v-t graph

5.2 Velocity vs. TimeX-axis is the timeY-axis is the velocitySlope of the line = the acceleration

Different Velocity-time graphs

Different Velocity-time graphs

Velocity vs. TimeHorizontal lines = constant velocitySloped line = changing velocitySteeper = greater change in velocity per secondNegative slope = deceleration

Acceleration vs. TimeTime is on the x-axisAcceleration is on the y-axisShows how acceleration changes over a period of time.Often a horizontal line.

All 3 Graphsvtat

Real lifeatvtNote how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the orange segments would be connected. In our class, however, well only deal with constant acceleration.

Constant Rightward Velocity

Constant Leftward Velocity

Constant Rightward Acceleration

Constant Leftward Acceleration

Leftward Velocity with Rightward Acceleration

Graph PracticeTry making all three graphs for the following scenario:1. Newberry starts out north of home. At time zero hes driving a cement mixer south very fast at a constant speed. 2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose.3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail.4. Fleeing the scene of the crime, Newberry takes off again 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.

Area Underneath v-t GraphIf you calculate the area underneath a v-t graph, you would multiply height X width.Because height is actually velocity and width is actually time, area underneath the graph is equal to Velocity X time or Vt

Remember that Velocity = d tRearranging, we get d = velocity X t

So.the area underneath a velocity-time graph is equal to the displacement during that time period.

AreaNote that, here, the areas are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this as well.

Velocity vs. TimeThe area under a velocity time graph indicates the displacement during that time period.Remember that the slope of the line indicates the acceleration.The smaller the time units the more instantaneous is the acceleration at that particular time.If velocity is not uniform, or is changing, the acceleration will be changing, and cannot be determined for an instant, so you can only find average acceleration

5.3 AccelerationDetermine from the curves on a velocity-time graph both the constant and instantaneous accelerationDetermine the sign of acceleration using a v-t graph and a motion diagramCalculate the velocity and the displacement of an object undergoing constant acceleration

5.3 AccelerationLike speed or velocity, acceleration is a rate of change, defined as the rate of change of velocityAverage Acceleration = change in velocity

Elapsed timeUnits of acceleration?

Rearrangement of the equation

Finally

This equation is to be used to find (final) velocity of an accelerating object. You can use it if there is or is not a beginning velocity

Displacement under Constant AccelerationRemember that displacement under constant velocity wasWith acceleration, there is no one single instantaneous v to use, but we could use an average velocity of an accelerating object.

d = vt or d1 = d0 + vt

Average velocity of an accelerating objectV = (v0 + v1)Average velocity of an accelerating object would simply be of sum of beginning and ending velocities

So.Key equation

Some other equations2This equation is to be used to find final position when there is an initial velocity, but velocity at time t1 is not known.

If no time is known, use this to find final position.22

Finding final velocity if no time is known22

The equations of importance

22222

Practical Application Velocity/Position/Time equationsCalculation of arrival times/schedules of aircraft/trains (including vectors)GPS technology (arrival time of signal/distance to satellite)Military targeting/deliveryCalculation of Mass movement (glaciers/faults)Ultrasound (speed of sound) (babies/concrete/metals) Sonar (Sound Navigation and Ranging)Auto accident reconstructionExplosives (rate of burn/expansion rates/timing with det. cord)

5.4 Free FallRecognize the meaning of the acceleration due to gravityDefine the magnitude of the acceleration due to gravity as a positive quantity and determine the sign of the acceleration relative to the chosen coordinate systemUse the motion equations to solve problems involving freely falling objects

FreefallDefined as the motion of an object if the only force acting on it is gravity.No friction, no air resistance, no drag

Acceleration Due to GravityGalileo Galilei recognized about 400 years ago that, to understand the motion of falling objects, the effects of air or water would have to be ignored.As a result, we will investigate falling, but only as a result of one force, gravity.Galileo Galilei 1564-1642

Galileos RampsBecause gravity causes objects to move very fast, and because the time-keepers available to Galileo were limited, Galileo used ramps with moveable bells to slow down falling objects for accurate timing.

Galileos Ramps

Galileos RampsTo keep accurate time, Galileo used a water clock.For the measurement of time, he employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which he collected in a small glass during the time of each descent... the water thus collected was weighed, after each descent, on a very accurate balance; the difference and ratios of these weights gave us the differences and ratios of the times...

Displacements of Falling ObjectsLooking at his results, Galileo realized that a falling ( or rolling downhill) object would have displacements that increased as a function of the square of the time, or t2Another way to look at it, the velocity of an object increased as a function of the square of time, multiplied by some constant.

Galileo also found that all objects, no matter what slope of ramp he rolled them down, and as long as the ramps were all the same height, would reach the bottom with the same velocity.

Galileos FindingGalileo found that, neglecting friction, all freely falling objects have the same acceleration.

Hippo & Ping Pong Ball In a vacuum, all bodies fall at the same rate.When theres no air resistance, size and shape matter not!If a hippo and a ping pong ball were dropped from a helicopter in a vacuum (assuming the copter could fly without air), theyd land at the same time.

Proving Galileo CorrectGalileo could not produce a vacuum to prove his ideas. That came later with the invention of a vacuum machine, and the demonstration with a guinea feather and gold coin dropped in a vacuum.

Guinea Feather and Coin/NASA demonstrations

Acceleration Due to GravityGalileo calculated that all freely falling objects accelerate at a rate of9.8 m/s2This value, as an acceleration, is known as g

Acceleration Due to GravityBecause this value is an acceleration value, it can be used to calculate displacements or velocities using the acceleration equations learned earlier. Just substitute g for the a

Example problemA brick is dropped from a high building. What is its velocity after 4.0 sec.?How far does the brick fall during this time?

The Churchs opposition to new thoughtChurch leaders of the time held the same views as Aristotle, the great philosopher.Aristotle thought that objects of different mass would fall at different ratesmakes sense huh?????? All objects have their natural position. Rocks fall faster than feathers, so it only made sense (to him)Galileo, in attempting to convince church leaders that the natural position view was incorrect, considered two rocks of different mass.

Falling Rock ConundrumGalileo presented this in his book Dialogue Concerning the Two Chief World Systems(1632) as a discussion between Simplicio (as played by a church leader) and Salviati (as played by Galileo)Two rocks of different masses are dropped Massive rock falls faster???

Rocks continuedNow consider the two rocks held together by a length of string.If the smaller rock were to fall slower, it would impede the rate at which both rocks would fall. But the two rocks together would actually have more mass, and should therefore fall faster.A conundrum????? The previously held views could not have been correct.

Galileo had data which proved Aristotle and the church wrong, but church leaders were hardly moved in their position that all objects have their correct position in the worldFurthermore, the use of Simplicio (or simpleton) as the head of the church in his dialog, was a direct insult to the church leaders themselves.