Chapter 3. Motion in One Dimension

Size: px

Start display at page:

Download "Chapter 3. Motion in One Dimension"

Dwight Cox
5 years ago
Views:

1 Chapter 3 Motion in One Dimension

2 Outline 3.1 Position, Velocity and Speed 3.2 Instantaneous Velocity and Speed 3.3 Acceleration 3.4 Motion Diagrams 3.5 One-Dimensional Motion with Constant Acceleration 3.6 Freely Falling Objects

3 3.1 Position, Velocity and Speed Position An object s position is its location with respect to a chosen reference point Displacement the change in position during some time interval represented by x x x f x i x is positive when x f > x i x is negative when x f < x i

4 Vectors and Scalars A vector physical quantity that require the specification of both direction and magnitude use + and - sign to indicate vector directions ex : displacement, velocity and acceleration Displacement in positive (+ x) means object move to the right and (- x) means move to left A scalar A quantity that has magnitude but no direction ex : position, speed and etc.

5 3.1 Position, Velocity and Speed cont. Average speed Speed is a scalar quantity Defined as Average velocity the displacement x occurs at the interval of time. x indicates motion along x- axis It has no direction, so always positive value. Average velocity of a moving particle can be positive and negative depends on the sign of displacement x is positive when x f > x i, V x. avg is positive x is negative when x f < x i,, V x. avg is negative

6 3.1 Position, Velocity and Speed cont. Average speed Distance traveled per unit of time Average velocity Displacement traveled per unit of time d = 30 m B d = 30 m x =12 m B A A 20 o Time t = 5 s Time t = 5 s Direction is not (scalar) V = 2.4 m/s at 20 0 N of E Direction required (vector)

7 3.2 Instantaneous Velocity and Speed Instantaneous velocity can indicate what is happening at every point of time. General equation for instantaneous velocity : v x x lim0 t t dt Instantaneous velocity can be positive, negative or zero. Instantaneous speed is the magnitude of the instantaneous velocity but it has no direction. dx

8 Instantaneous Velocity, graph The instantaneous velocity is the slope of the line tangent to the x vs. t curve. This would be the green line. The light blue lines show that as t gets smaller, they approach the green line.

9 3.3 Acceleration Acceleration is the velocity changes with time. The average acceleration is the change in velocity V x divided by the time interval t during the change occurred. + and - sign can indicate direction. a x, avg v v v t t t x xf xi f i Dimension : L/T 2 SI : m /s 2

10 Instantaneous Acceleration the limit of the average acceleration as t approaches 0. is also equals to the derivative of the velocity with respect to time. acceleration is the slope of Vx vs t graph. a x lim x x t 0 t dt dt 2 v dv dx 2 With : Vx = dx/dt It has positive and negative value to indicate the direction.

11 Instantaneous Acceleration graph The slope of the velocity-time graph is the acceleration. The green line represents the instantaneous acceleration. The blue line is the average acceleration.

= ( 40 5t 2 ) m/s, where t is in seconds.

12 Ex 3.4 Average and Instaneous Acceleration The velocity of a particle moving along the x axis varies in time according to the expression V x = ( 40 5t 2 ) m/s, where t is in seconds. a) Find the average acceleration in the time interval t=0 to t=2s b) Determine acceleration at t=2s

13 Notes about Velocity and Acceleration Negative acceleration does not necessarily mean the object is slowing down. If the acceleration and velocity are in the same direction no matter + / -, the object is speeding up. If the acceleration and velocity are in the opposite direction, the object is slowing down.

14 3.4 Motion Diagram A motion diagram can be formed by imagining the stroboscope photograph of a moving object. Red arrows represent velocity. Purple arrows represent acceleration. All purple arrows maintain the same length means acceleration are constant.

15 Constant Velocity The car image are equally spaced. The car is moving with constant positive velocity. (red arrows remain the same size) Acceleration equals to zero.

Acceleration same direction with Velocity The car image farther apart as time increases Acceleration and velocity are in the same direction Velocity is

16 Acceleration same direction with Velocity The car image farther apart as time increases Acceleration and velocity are in the same direction Velocity is increasing as the arrows are getting longer Acceleration is constant as the arrows maintain the same length Thus, it is positive acceleration and positive velocity

Acceleration opposite direction with Velocity The car image become closer as time increases Acceleration and velocity are in the opposite direction Velocity is

17 Acceleration opposite direction with Velocity The car image become closer as time increases Acceleration and velocity are in the opposite direction Velocity is decreasing as the arrows are getting shorter Acceleration is constant as the arrows maintain the same length Thus, it is negative acceleration and positive velocity

18 3.5 One-dimensional Motion with Constant Acceleration Constant acceleration. Velocity changes at the same rate throughout the motion. Kinematic Equation 1 : v v a t xf xi x Can determine the velocity at any time t when we know its initial velocity and its acceleration Doesn t tell about displacement

19 Kinematic Equation 2: The average velocity in a time interval can be expressed as the arithmetic mean of the initial velocity V xi and final velocity V xf is only suit for constant acceleration. v x, avg v xi v 2 xf

20 Kinematic Equation 3: This gives us about the position of the object in terms of time and velocities. Doesn t tell acceleration Substitute Equation 2 then v x, avg v xi v 2 xf Thus,

21 Kinematic Equation 4: This gives final position in terms of velocity and acceleration. Doesn t tell final velocity Substitute Equation 3 with Equation 1 Equation 3 : Equation 1: v v a t xf xi x then 1 xf xi v xit axt 2 2

22 Kinematic Equation 5: Tells final velocity in terms of acceleration and displacement. Doesn t include with time Equation 1: v v a t xf xi x then Equation 3 : Substitute Equation 1 with Equation 3 v v 2a x x 2 2 xf xi x f i

23 When a = 0 When the acceleration is zero, v xf = v xi = v x x f = x i + v x t When acceleration is zero, velocity is constant and displacement changes linearly with time.

24 Kinematic Equations - Summary

25 Example (3.7) : Carrier Landing A jet lands on an aircraft carrier at 140 mi/h ( 63 m/s). (a) What is its acceleration if it stops in 2.0 s? (b) What is the displacement of the plane while it is stopping? Example (3.8) : Watch Out for the Speed Limit! A car traveling at a constant speed of 45.0 m/s passes a trooper hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets out from the billboard to catch it, accelerating at a constant rate of 3.00 m/s 2. How long does it take her to overtake the car?

26 3.6 Freely Falling Objects A freely falling objects is any object that moves freely under the influence of gravity alone It doesn t depend upon the initial motion of the object. Thrown upward Thrown downward Dropped (release from rest) Any freely falling object experiences an acceleration directed downward, regardless of the initial motion.

27 3.6 Freely Falling Objects cont. The magnitude of free fall acceleration is symbol g g decreases with increasing altitude g varies with latitude Define up as +y direction the g value at Earth s surface is 9.80 m/s 2 If we neglect air resistance, assume acceleration doesn t vary with altitude over distance, the free fall motion is constantly accelerated motion in one dimension. While the motion in vertical direction and acceleration is downward we always take ay = -g = m/s2, the minus sign shows the acceleration of a freely falling object is downward.

28 Free Fall An Object Dropped Initial velocity, v o = 0 Assume upward direction is positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is a y = -g = m/s 2 v o = 0 a = -g

29 Free Fall An Object Thrown Downward a y = -g = m/s 2 Initial velocity 0 With upward being positive, initial velocity will be negative. v o 0 a = -g Section 2.7

30 Free Fall Object Thrown Upward Initial velocity is upward, so positive At the maximum height, The instantaneous velocity is zero. a y = -g = m/s 2 everywhere in the motion v = 0 v o 0 a = -g

UP = + Release Point a = - y = + v = 0 v = + v = - y = + y = + v = - y = 0 6.

31 UP = + Release Point a = - y = + v = 0 v = + v = - y = + y = + v = - y = Sign Convention: A Ball Thrown Vertically Upward Displacement is positive (+) or negative (-) based on LOCATION. Velocity is positive (+) or negative (-) based on direction of motion. a = - y = - v= - Tippens Acceleration is (+) or (-) based on direction of force (weight).

Example (3.10) : Not a bad Throw for a Rookie! A stone thrown from the top of a building is given an initial velocity of 20.0 m/s straight upward. The building is 50.

32 Example (3.10) : Not a bad Throw for a Rookie! A stone thrown from the top of a building is given an initial velocity of 20.0 m/s straight upward. The building is 50.0 m high, and the stone just misses the edge of the roof on its way down, as shown in Figure (2.14). Using t A = 0 as the time the stone leaves the thrower s hand at position, determine : (a) The time at which the stone reaches its maximum height, (b) The maximum height, (c) The time at which the stone returns to the height from which it was thrown, (d) (e) The velocity of the stone at this instant, and The velocity and position of the stone at t = 5.00 s

Chapter 2. Motion in One Dimension

Chapter 2. Motion in One Dimension Chapter 2 Motion in One Dimension Types of Motion Translational An example is a car traveling on a highway. Rotational An example is the Earth s spin on its axis. Vibrational An example is the back-and-forth