Presentation on theme: "Motion in One Dimension"— Presentation transcript:
1 Motion in One Dimension Chapter 2Motion in One Dimension
2 KinematicsTo describe motion while ignoring the agents that caused the motionFor now, we will consider motion in one dimension - along a straight line -, and use the particle model.A particle is a point-like object, has mass but infinitesimal size
3 Position Defined in terms of a frame of reference The reference frame must has an origin.The object’s position is its location with respect to the origin of the frame of reference.
4 Position-Time GraphThe position-time graph shows the motion of the particle (car)The smooth curve is a guess as to what happened between the data points
5 DisplacementDefined as the change in position during some time interval DtRepresented as xXi is the initial position and Xf is the final position.Different than distance – the length of a path followed by a particle
6 Average VelocityThe average velocity is rate at which the displacement occursThe dimensions are length / time [L/T]The slope of the line connecting the initial and final points in the position – time graph
7 Average Velocity, cont Gives no details about the motion Gives the result of the motionIt can be positive or negativeIt depends on the sign of the displacementIt can be interpreted graphicallyIt will be the slope of the position-time graph
8 Displacement from a Velocity - Time Graph The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve
15 Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zeroThe instantaneous velocity indicates what is happening at every point of time
16 Instantaneous Velocity, graph The instantaneous velocity is the slope of the line tangent to the x vs. t curveThis would be the green lineThe blue lines show that as t gets smaller, they approach the green line
17 Instantaneous Velocity, equations The general equation for instantaneous velocity isWhen “velocity” is used in the text, it will mean the instantaneous velocity
18 Instantaneous Velocity, Signs At point A, vx is positiveThe slope is positiveAt point B, vx is zeroThe slope is zeroAt point C, vx is negativeThe slope is negative
19 Instantaneous SpeedThe instantaneous speed is the magnitude of the instantaneous velocity vectorSpeed can never be negative
30 2.3 Constant Velocity If the velocity of a particle is constant Its instantaneous velocity at any instant is the same as the average velocity over a given time periodvx = v x avg = Dx / DtAlso, xf = xi + vx tThese equations can be applied to particles or objects that can be modeled as a particle moving under constant velocity
32 2.4 Average AccelerationAcceleration is the rate of change of the velocityIt is a measure of how rapidly the velocity is changingDimensions are L/T2SI units are m/s2
33 Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as t approaches 0
34 Instantaneous Acceleration – Graph The slope of the velocity vs. time graph is the accelerationPositive values correspond to where velocity in the positive x direction is increasingThe acceleration is negative when the velocity is in the positive x direction and is decreasingFig 2.7
42 2.5 Negative Acceleration Negative acceleration does not necessarily mean that an object is slowing downIf the velocity is negative and the acceleration is negative, the object is speeding upDeceleration is commonly used to indicate an object is slowing down, but will not be used in this text
43 Acceleration and Velocity, 1 When an object’s velocity and acceleration are in the same direction, the object is speeding up in that directionWhen an object’s velocity and acceleration are in the opposite direction, the object is slowing down
44 Acceleration and Velocity, 2 The car is moving with constant positive velocity (shown by red arrows maintaining the same size)Acceleration equals zero
45 Acceleration and Velocity, 3 Velocity and acceleration are in the same directionAcceleration is uniform (blue arrows maintain the same length)Velocity is increasing (red arrows are getting longer)This shows positive acceleration and positive velocity
46 Acceleration and Velocity, 4 Acceleration and velocity are in opposite directionsAcceleration is uniform (blue arrows maintain the same length)Velocity is decreasing (red arrows are getting shorter)Positive velocity and negative acceleration
48 Kinematic Equations, specific For constant a,Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration
49 Kinematic Equations, specific For constant acceleration,The average velocity can be expressed as the arithmetic mean of the initial and final velocitiesOnly valid when acceleration is constant
50 Kinematic Equations, specific For constant acceleration,This formula gives the position of the particle in terms of the initial and final velocitiesThe acceleration is not given.
51 Kinematic Equations, specific As constant acceleration is given,Gives final position in terms of initial velocity and accelerationThe final velocity is not necessary in this formula.
52 Kinematic Equations, specific For constant accelation,The final velocity is determined by initial velocity, acceleration and displacementThere is no information about the time interval of the traveling.
71 2.7 Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity aloneIt does not depend upon the initial motion of the objectDropped – released from restThrown downwardThrown upward
72 Acceleration of Freely Falling Object The acceleration of an object in free fall is directed downward, regardless of the initial motionThe magnitude of free fall acceleration is g = 9.80 m/s2g decreases with increasing altitudeg varies with latitude9.80 m/s2 is the average at the earth’s surface
73 Acceleration of Free Fall, cont. We will neglect air resistanceFree fall motion is constantly accelerated motion in one dimensionLet upward be positiveUse the kinematic equations with ay = g = m/s2The negative sign indicates the direction of the acceleration is downward
77 Free Fall ExampleInitial velocity at A is upward (+) and acceleration is g (-9.8 m/s2)At B, the velocity is 0 and the acceleration is g (-9.8 m/s2)At C, the velocity has the same magnitude as at A, but is in the opposite directionThe displacement is –50.0 m (it ends up 50.0 m below its starting point)