Presentation on theme: "Motion in One Dimension"— Presentation transcript:
1 Motion in One Dimension Chapter 2Motion in One Dimension
2 2.1 Position, Velocity, and Speed 2.2 Instantaneous Velocity and Speed2.3 Acceleration2.4 Freely Falling Objects2.5 Kinematic Equations Derived from Calculus.
3 KinematicsKinematics describes motion while ignoring the agents that caused the motionFor now, we will consider motion in one dimensionAlong a straight lineWe will use the particle modelA particle is a point-like object, that has mass but infinitesimal size
4 Position Position is defined in terms of a frame of reference For one dimension the motion is generally along the x- or y-axisThe object’s position is its location with respect to the frame of reference
5 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
6 DisplacementDisplacement is defined as the change in position during some time intervalRepresented as xx = xf - xiSI units are meters (m), x can be positive or negativeDisplacement is different than distance. Distance is the length of a path followed by a particle.
7 Vectors and ScalarsVector quantities that need both -magnitude (size or numerical value) and direction to completely describe themWe will use + and – signs to indicate vector directionsScalar quantities are completely described by magnitude only
8 Average VelocityThe average velocity is the rate at which the displacement occursThe dimensions are length / time [L/T]The SI units are m/sIs also the slope of the line in the position – time graph
9 Average Speed Speed is a scalar quantity same units as velocitytotal distance / total timeThe average speed is not (necessarily) the magnitude of the average velocity
10 Instantaneous Velocity Instantaneous velocity is 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
11 Instantaneous Velocity The general equation for instantaneous velocity isThe instantaneous velocity can be positive, negative, or zero
12 Instantaneous Velocity 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
13 Instantaneous SpeedThe instantaneous speed is the magnitude of the instantaneous velocityRemember that the average speed is not the magnitude of the average velocity
14 Average AccelerationAcceleration is the rate of change of the velocityDimensions are L/T2SI units are m/s²
15 Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as t approaches 0
16 Instantaneous Acceleration The slope of the velocity vs. time graph is the accelerationThe green line represents the instantaneous accelerationThe blue line is the average acceleration
17 Acceleration and Velocity When an object’s velocity and acceleration are in the same direction, the object is speeding upWhen an object’s velocity and acceleration are in the opposite direction, the object is slowing down
18 Acceleration and Velocity The car is moving with constant positive velocity (shown by red arrows maintaining the same size)Acceleration equals zero
19 Acceleration and Velocity 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
20 Acceleration and Velocity 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
21 1D motion with constant acceleration tf – ti = t
22 1D motion with constant acceleration In a similar manner we can rewrite equation for average velocity:and than solve it for xfRearranging, and assuming
23 1D motion with constant acceleration (1)Usingand than substituting into equation for final position yields(1)(2)(2)Equations (1) and (2) are the basic kinematics equations
24 1D motion with constant acceleration These two equations can be combined to yield additional equations.We can eliminate t to obtainSecond, we can eliminate the acceleration a to produce an equation in which acceleration does not appear:
25 Kinematics with constant acceleration - Summary
27 Kinematic EquationsThe kinematic equations may be used to solve any problem involving one-dimensional motion with a constant accelerationYou may need to use two of the equations to solve one problemMany times there is more than one way to solve a problem
28 Kinematics - Example 1How long does it take for a train to come to rest if it decelerates at 2.0m/s2 from an initial velocity of 60 km/h?
29 A car is approaching a hill at 30 A car is approaching a hill at 30.0 m/s when its engine suddenly fails just at the bottom of the hill. The car moves with a constant acceleration of –2.00 m/s2 while coasting up the hill. (a) Write equations for the position along the slope and for the velocity as functions of time, taking x = 0 at the bottom of the hill, where vi = 30.0 m/s. (b) Determine the maximum distance the car rolls up the hill.
30 Graphical Look at Motion: displacement-time curve The slope of the curve is the velocityThe curved line indicates the velocity is changingTherefore, there is an acceleration
31 Graphical Look at Motion: velocity-time curve The slope gives the accelerationThe straight line indicates a constant acceleration
32 Graphical Look at Motion: acceleration-time curve The zero slope indicates a constant acceleration
33 Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity alone.It does not depend upon the initial motion of the objectDropped – released from restThrown downwardThrown upward
34 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
35 Acceleration of Free Fall We will neglect air resistanceFree fall motion is constantly accelerated motion in one dimensionLet upward be positiveUse the kinematic equations with ay = g = m/s2
36 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 direction
37 A student throws a set of keys vertically upward to her sorority sister, who is in a window 4.00 m above. The keys are caught 1.50 s later by the sister's outstretched hand. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught?
38 A ball is dropped from rest from a height h above the ground A ball is dropped from rest from a height h above the ground. Another ball is thrown vertically upwards from the ground at the instant the first ball is released. Determine the speed of the second ball if the two balls are to meet at a height h/2 above the ground.
39 A freely falling object requires 1. 50 s to travel the last 30 A freely falling object requires 1.50 s to travel the last 30.0 m before it hits the ground. From what height above the ground did it fall?
40 Motion Equations from Calculus Displacement equals the area under the velocity – time curveThe limit of the sum is a definite integral
42 Kinematic Equations – Calculus Form with Constant Acceleration The integration form of vf – vi givesThe integration form of xf – xi gives
43 The height of a helicopter above the ground is given by h = 3 The height of a helicopter above the ground is given by h = 3.00t3, where h is in meters and t is in seconds. After 2.00 s, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground?
44 Automotive engineers refer to the time rate of change of acceleration as the "jerk." If an object moves in one dimension such that its jerk J is constant, (a) determine expressions for its acceleration ax(t), velocity vx(t), and position x(t), given that its initial acceleration, speed, and position are axi , vxi, and xi , respectively. (b) Show that
45 The acceleration of a marble in a certain fluid is proportional to the speed of the marble squared, and is given (in SI units) by a = –3.00 v2 for v > 0. If the marble enters this fluid with a speed of 1.50 m/s, how long will it take before the marble's speed is reduced to half of its initial value?
46 A test rocket is fired vertically upward from a well A test rocket is fired vertically upward from a well. A catapult gives it initial velocity 80.0 m/s at ground level. Its engines then fire and it accelerates upward at 4.00 m/s2 until it reaches an altitude of m. At that point its engines fail and the rocket goes into free fall, with an acceleration of –9.80 m/s2. (a) How long is the rocket in motion above the ground? (b) What is its maximum altitude? (c) What is its velocity just before it collides with the Earth?
47 An inquisitive physics student and mountain climber climbs a 50 An inquisitive physics student and mountain climber climbs a 50.0-m cliff that overhangs a calm pool of water. He throws two stones vertically downward, 1.00 s apart, and observes that they cause a single splash. The first stone has an initial speed of 2.00 m/s. (a) How long after release of the first stone do the two stones hit the water? (b) What initial velocity must the second stone have if they are to hit simultaneously? (c) What is the speed of each stone at the instant the two hit the water?