Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion, without consideration of what causes the motion. Dynamics

Kinematics in One Dimension

Kinematics deals with the concepts that are needed to describe motion, without consideration of what causes the motion.

Dynamics deals with the effect that forces have on motion. Analyzes the causes of motion.

Together, kinematics and dynamics formthe branch of physics known as Mechanics.

MechanicsBranch of Physics

Kinematics Study of motion

Dynamics What produces and effects motion

What is Motion?Motion

Object is in motion when it is changing position with respect to the surroundingsExample: Car driving down a road

Kinematics in One DimensionConcerned with the description of an object that moves along a straight-line path Linear motionConvenient to specify motion along the x and y coordinate systemOrient the straight line path along the coordinate axes so motion is along one of the axes

Kinematics in One DimensionImportant to specify not only speed but also the direction of motion (Up or down; North, South, East, or West)Coordinate (x and y) axesObjects to the right of the origin on the x axis = positiveObjects to the left of the origin on the x axis = negativeObjects above the origin on the y axis = positiveObjects below the origin on the y axis = negative

Displacement versus DistanceDisplacement (x)Change in position of the objectHow far the object is from its starting point

Displacement may not equal total distance traveled

DistanceDistanceTotal path length traversed in moving from one location to anotherExample: Jimmy is driving to school but he forgets to pick up Johnny on the wayHe now has to reverse his direction and drive back 2 milesTotal Distance Traveled = 12 milesTotal Displacement = 8 miles+ 10 miles - 2 miles

Displacement versus DistancePerson walks 70m East then turns around and walks 30m West Total distance traveled = 100m Displacement = 40m, since the person is now only 40m from the starting point

** Displacement: Quantity with both magnitude and direction Vector* Represented by arrows in diagrams* Displacement = Final position Initial position

Displacement

Speed versus VelocitySpeed and velocity

Used synonymously in everyday conversation But, the terms have different meanings in physics

Speed and VelocityAverage speed is the total distance traveled divided by the time required to travel this distance.SI units for speed: meters per second (m/s)

Speed and VelocityExample: Distance Run by a Jogger

How far does a jogger run in 1.5 hours (5400 s) if her average speed is 2.22 m/s?

Answer

Distance traveled = (Average speed) (Elapsed Time)

Distance= (2.22 m/s)(5400 s)= 12,000 m

Speed and VelocityAverage velocity is the displacement divided by the elapsed time.

Average VelocityThe football player ran 50 m in 20 s. What is his average velocity?

v = 50 m / 20 s = 2.5 m/s

Speed and VelocityExample: The Worlds Fastest Jet-Engine Car

Andy Green in the car ThrustSSC set a world record of 341.1 m/s (762.8 mph) in 1997. To establish such a record, the driver makes two runs through the course, one in each direction,to nullify wind effects. From the data, determine the averagevelocity for each run.

Speed and Velocity(759 mph)(766.4 mph)

Speed versus VelocitySpeed: Positive number, with unitsDoes not take into account directionSpeed is therefore a _ _ _ _ _ _ quantity?Velocity (v):Specifies both the magnitude (numerical value how fast an object is moving) and also the direction in which an object is movingVelocity is therefore a _ _ _ _ _ _ quantity?

Instantaneous VelocityYou drive a car 100 km along a straight road in one direction for 4 hoursAverage velocity = 25 km/hrunlikely you were driving 25 km/hr at every instant

Instantaneous velocity Average velocity at any instant of timeExample: Speedometer readingGives the speed of a car over a very short interval of time

Speed and VelocityThe instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.Animation of Average vs. Instantaneous Speed

Graphical Analysis -Uniform Linear MotionGiven below is a strobe picture of aball rolling across a tableStrobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 secondsNotice that the ball covers an equal distance between flashesAssume distance between flashes = 20 cm Display the ball's behavior on an position versus time graph

Position versus Time Graph

Slope of a Straight LineSlope of a straight lineSlope =

On the position versus time graphSlope of the straight line = Average velocityPositive slope indicates that the objects displacement, away from the origin, increases with time so, motion of the ball is in the positive x direction

Slope of a Position versus Time Graph

What does the Slope of a Position vs. Time Graph Tell Us?On the Position vs. Time GraphSlope of the straight line 20 cm/0.1 s = 200 cm/sec Represents the balls average velocity, as it moves across the tableBalls velocity is positive, therefore it is moving in the positive directionBall's velocity Vector quantity possessing both magnitude (200 cm/sec) and direction (positive)

Velocity versus Time GraphSlope of the straight line on the Position versus Time Graph Average VelocityVelocityTimeAverage Velocity = 200 cm/s200 cm/s

What would the Acceleration versus Time Graph look like?Hint: Slope of the straight line on the Velocity versus Time Graph Average AccelerationAccelerationTime

Acceleration versus Time Graph* Slope of the straight line on the Velocity versus Time Graph = 0** Average Acceleration = 0*** Ball is moving with constant velocityAccelerationTime

AccelerationAverage Acceleration (a) Describes how rapidly the velocity of an object is changingDefined as the change in velocity divided by the time taken to make this changeCar velocity increases in magnitude from zero to 70 mph acceleration

Constant Acceleration

AccelerationChange in velocity per time interval Velocity = m/sTime = s Acceleration formula leads to units of m/s2

Animation of constant acceleration website

AccelerationNotion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

AccelerationDEFINITION OF AVERAGE ACCELERATION

AccelerationExample: Acceleration and Increasing Velocity

Determine the average acceleration of the plane.

Acceleration

AccelerationExample Acceleration and Decreasing Velocity

Acceleration

AccelerationAcceleration:Vector quantity For one-dimensional motion:Utilize a (+) positive sign or (-) negative sign to indicate direction relative to a coordinate system (x or y axes) SI units for acceleration:meters/second2 m/s2

Acceleration Animation Website

Animation - Direction of Acceleration and Velocity Website

Motion at Constant Acceleration

Constant Acceleration:Instantaneous and average accelerations are equal

Utilize a set of kinematic equations for constant acceleration

Equations of Kinematics for Constant AccelerationIt is customary to dispense with the use of boldface symbols overdrawn with arrows for the displacement, velocity, and acceleration vectors. We will, however, continue to convey the directions with a plus or minus sign.

Equations of Kinematics for Constant AccelerationLet the object be at the origin when the clock starts.

Equations of Kinematics for Constant Acceleration

Equations of Kinematics for Constant AccelerationFive kinematic variables:

1. Displacement x

2. Acceleration (constant) a

3. Final velocity (at time t) v

4. Initial velocity vo

5. Elapsed time t

Equations of Kinematics for Constant AccelerationExample: Catapulting a Jet

** Find its displacement.

Applications of the Equations of KinematicsSolving ProblemsDraw a diagram of the situationIllustrate coordinate axes decide which directions are to be called positive (+) and negative (-)Write down the known quantities for any of the five kinematic variablesVerify that the information contains values for at least three of the five kinematic variables. Select the appropriate equationWhen the motion is divided into segments, remember the final velocity of one segment is the initial velocity for the nextCarry out the calculation, keeping track of the unitsIs the result reasonable?

Applications of the Equations of KinematicsExample: An Accelerating Spacecraft

A spacecraft is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing?

xavvot+215000 m-10.0 m/s2?+3250 m/s

Applications of the Equations of Kinematics

Applications of the Equations of Kinematics

xavvot+215000 m-10.0 m/s2?+3250 m/s

Falling ObjectsGalileo Galileis contribution to our understanding of the motion of falling objects ** In the absence of air resistance, at any given location on Earth, all objects fall with the same constant acceleration*** Acceleration due to gravityg = 9.8 m/s2Uniformly accelerated motion object allowed to fall freely near the Earths surface

Acceleration due to GravityVector

Direction is toward the center of the Earth

Freely Falling BodiesIn the absence of air resistance, it is found that all bodies at the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is small compared to the radius of the Earth, then the acceleration remains essentially constant throughout the descent.This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity.

Kinematic Equations:Constant AccelerationKinematic Equations: Acceleration due to Gravity

Motion is Relative!Using the Earth as a frame of reference most commonObject with a positive velocity (+ v) Moving to the right (Away from the reference point)Object with a negative velocity (- v) Moving to the left (In the opposite direction with respect to the reference point)Hint: It is possible for an object to have a zero velocity (v = 0), while it changes direction, in the presence of an acceleration* Just remember! It is impossible for acceleration to be zero and have a changing velocity

Things to Remember!Objects can either be at RestMoving with a constant velocity orAcceleratingIf an object is at rest or moving with a constant velocity Acceleration = 0!When present, we always assume accelerations are uniform, or constant

Freely Falling Bodies

Free-FallWebsite:Elephant and FeatherFree Fall Different meaning to a physicist than it does to a skydiverIn physicsFree fall = One-dimensional motion of any object under the influence of gravity only No air resistance or frictional effects of any kindWhereas it is air resistance that makes skydiving a hobby rather than a crazy stunt!

Free FallPhysics Free Fall Since everything we observe falling is falling through the air, is Physics Free Fall, a useless idea in practice?No! Any falling object's motion is at least approximately free fall as long as:It is relatively heavy compared to its sizeDropping a ball or jumping off a chair = Free-fall motionDropping an unfolded piece of paper, or the motion of a dust particle floating in the air Not Free Fall (If you crumble the paper into a "paper wad", however, its motion is approximately free fall)

Free FallIt falls for a relatively short timeJumping off a chair = Free fallAfter you have jumped out of an airplane and fallen for several seconds you are not in free fall, since air resistance is now a factor in your motionIt is moving relatively slowlyDrop a ball or throw it down Free fallShoot a ball of a cannon Not free fall* Note: Object does not have to be falling to be in free fall Example - Throwing a ball upward Motion is still considered to be free fall, since it is moving under the influence of gravity

Freely Falling BodiesExample: A Falling Stone

A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone?


yavvot?- 9.80 m/s20 m/s3.00 s

Freely Falling BodiesExample: How High Does it Go?

The referee tosses the coin upwith an initial speed of 5.00m/s.In the absence of air resistance,how high does the coin go aboveits point of release?


yavvot?- 9.80 m/s20 m/s+ 5.00 m/s


yavvot?-9.80 m/s20 m/s+5.00 m/s

Freely Falling BodiesConceptual Example: Acceleration Versus Velocity

There are three parts to the motion of the coin On the way up, the coin has a vector velocity that is directed upward and has decreasing magnitude At the top of its path, the coin momentarily has zero velocity On the way down, the coin has a downward-pointing velocity with an increasing magnitude.

** In the absence of air resistance, does the acceleration of the coin, like the velocity, change from one part to another?

Note: As this occurs, a = 9.81 m/s2 ! The object is changing direction at this point.

Freely Falling BodiesConceptual Example: Taking Advantage of Symmetry

Does the pellet in part b strike the ground beneath the cliff with a smaller, greater, or the same speed as the pellet in part a?

Analyzing GraphsThree types of graphs:

Three states of Motion:

Time TimeTimeDistanceVelocityAccelerationVelocity = 0 (rest)Constant velocityConstant acceleration

Graphical Analysis of Velocity and Acceleration

Velocity versus Time Graph?VelocityTime (s)4 m/s

Acceleration versus Time GraphAcceleration (m/s2)Time

Graphical Analysis of Velocity and Acceleration

On velocity vs. time graphs, the slope of a straight line is the acceleration of the object, and the area under the curve equals the distance the object travels.

WebsitesOne Dimensional KinematicsReview Lessons 1, 2, 3, 4, 5, 6* Disclaimer: This powerpoint presentation is a compilation of various works.

Example that illustrates distinction b/w total distance traveled and displacement Person walks 70m E then turns around and walks 30m W. Total distance traveled = 100m; Displacement = 40m, since the person is now only 40m from the starting pointDrives 10 miles- forgets to pick up his friend has to turn around and drive back 2 miles to pick up his friend Example Illustrates distinction b/w total distance traveled and displacementUtilizing the x axis to display the linear motionDisplacement = Final position Initial positionVelocity = Tells you how fast something is moving and in what directionWork on board1 meter = 3.28 feet; 1 mile = 5280 feetSpeed = scalar; Velocity = vectorLim delta t to zero = means we let delta t become extremely small approaching zeroStraight line represents constant velocityIllustrating how to derive kinematic equations from the definitions of velocity and acceleration**Evacuted tube no air resistance presentObjects are in motion solely under the influence of gravity = free fallNo- Page 34Conceptual ex 2-13*The ball has the same magnitude of velocity when it returns to the starting point as it did initially, but in the opposite direction (meaning of the negative sign)the motion is symmetrical about the maximum height (Page 34 + 35- Ex 2-14)*Represents- person moving away from origin (+ slope) at a constant speed (straight diagonal line)Represents- Person moving away from the origin (+ slope), at an increasing speed (straight diagonal line)*

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Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion, without consideration of what causes the motion. Dynamics