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Topic 4 : Uniform Acceleration

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Topic 4 : Uniform Acceleration Chapter 2 : Motion in One Dimension * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Ways an Object Might Accelerate The ... – PowerPoint PPT presentation

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Title: Topic 4 : Uniform Acceleration


1
Topic 4 Uniform Acceleration
  • Chapter 2 Motion in One Dimension

2
Quantities in Motion
  • Any motion involves three concepts
  • Displacement
  • Velocity
  • Acceleration
  • These concepts can be used to study objects in
    motion

3
2.1 Displacement
  • Defined as the change in position
  • f stands for final and i stands for initial
  • May be represented as ?y if vertical
  • Units are meters (m) in SI, centimeters (cm) in
    cgs or feet (ft) in US Customary

4
Displacements
5
Speed
  • The average speed of an object is defined as the
    total distance traveled divided by the total time
    elapsed
  • Speed is a scalar quantity

6
Speed, cont
  • Average speed totally ignores any variations in
    the objects actual motion during the trip
  • The total distance and the total time are all
    that is important
  • SI units are m/s

7
2.2 Velocity
  • It takes time for an object to undergo a
    displacement
  • The average velocity is rate at which the
    displacement occurs
  • generally use a time interval, so let ti 0

8
Speed vs. Velocity
  • Cars on both paths have the same average velocity
    since they had the same displacement in the same
    time interval
  • The car on the blue path will have a greater
    average speed since the distance it traveled is
    larger

9
Average Velocity, Constant
  • The straight line indicates constant velocity
  • The slope of the line is the value of the average
    velocity

10
Average Velocity, Non Constant
  • The motion is non-constant velocity
  • The average velocity is the slope of the blue
    line joining two points

11
Uniform Velocity
  • Uniform velocity is constant velocity
  • The instantaneous velocities are always the same
  • All the instantaneous velocities will also equal
    the average velocity

12
2.3 Acceleration
  • Changing velocity (non-uniform) means an
    acceleration is present
  • Acceleration is the rate of change of the
    velocity
  • Units are m/s² (SI), cm/s² (cgs), and ft/s² (US
    Cust)

13
Average Acceleration
  • Vector quantity
  • When the sign of the velocity and the
    acceleration are the same (either positive or
    negative), then the speed is increasing
  • When the sign of the velocity and the
    acceleration are in the opposite directions, the
    speed is decreasing

14
Average Acceleration
15
2.4 Relationship Between Acceleration and Velocity
  • Uniform velocity (shown by red arrows maintaining
    the same size)
  • Acceleration equals zero

16
Relationship Between Velocity and Acceleration
  • Velocity and acceleration are in the same
    direction
  • Acceleration is uniform (blue arrows maintain the
    same length)
  • Velocity is increasing (red arrows are getting
    longer)
  • Positive velocity and positive acceleration

17
Relationship Between Velocity and Acceleration
  • Acceleration and velocity are in opposite
    directions
  • Acceleration is uniform (blue arrows maintain the
    same length)
  • Velocity is decreasing (red arrows are getting
    shorter)
  • Velocity is positive and acceleration is negative

18
2.5 Kinematic Equations
  • Used in situations with uniform acceleration

19
Graphical Interpretation of the Equation
20
Problem-Solving Hints
  • Read the problem
  • Draw a diagram
  • Choose a coordinate system, label initial and
    final points, indicate a positive direction for
    velocities and accelerations
  • Label all quantities, be sure all the units are
    consistent
  • Convert if necessary
  • Choose the appropriate kinematic equation

21
Problem-Solving Hints, cont
  • Solve for the unknowns
  • You may have to solve two equations for two
    unknowns
  • Check your results
  • Estimate and compare
  • Check units

22
2.6 Free Fall
  • All objects moving under the influence of gravity
    only are said to be in free fall
  • Free fall does not depend on the objects
    original motion
  • All objects falling near the earths surface fall
    with a constant acceleration
  • The acceleration is called the acceleration due
    to gravity, and indicated by g

23
Acceleration due to Gravity
  • Symbolized by g
  • g 9.80 m/s²
  • When estimating, use g 10 m/s2
  • g is always directed downward
  • toward the center of the earth
  • Ignoring air resistance and assuming g doesnt
    vary with altitude over short vertical distances,
    free fall is constantly accelerated motion

24
Free Fall an object dropped
  • Initial velocity is zero
  • Let up be positive
  • Use the kinematic equations
  • Generally use y instead of x since vertical
  • Acceleration is g -9.80 m/s2

vo 0 a g
25
Free Fall an object thrown downward
  • a g -9.80 m/s2
  • Initial velocity ? 0
  • With upward being positive, initial velocity will
    be negative

26
Free Fall -- object thrown upward
  • Initial velocity is upward, so positive
  • The instantaneous velocity at the maximum height
    is zero
  • a g -9.80 m/s2 everywhere in the motion

v 0
27
Thrown upward, cont.
  • The motion may be symmetrical
  • Then tup tdown
  • Then v -vo
  • The motion may not be symmetrical
  • Break the motion into various parts
  • Generally up and down

28
Non-symmetrical Free Fall
  • Need to divide the motion into segments
  • Possibilities include
  • Upward and downward portions
  • The symmetrical portion back to the release point
    and then the non-symmetrical portion

29
3.3 Motion in Two Dimensions
  • Using or signs is not always sufficient to
    fully describe motion in more than one dimension
  • Vectors can be used to more fully describe motion
  • Still interested in displacement, velocity, and
    acceleration

30
Displacement
  • The position of an object is described by its
    position vector,
  • The displacement of the object is defined as the
    change in its position

31
Velocity
  • The average velocity is the ratio of the
    displacement to the time interval for the
    displacement
  • The instantaneous velocity is the limit of the
    average velocity as ?t approaches zero
  • The direction of the instantaneous velocity is
    along a line that is tangent to the path of the
    particle and in the direction of motion

32
Acceleration
  • The average acceleration is defined as the rate
    at which the velocity changes
  • The instantaneous acceleration is the limit of
    the average acceleration as ?t approaches zero

33
Unit Summary (SI)
  • Displacement
  • m
  • Average velocity and instantaneous velocity
  • m/s
  • Average acceleration and instantaneous
    acceleration
  • m/s2

34
Ways an Object Might Accelerate
  • The magnitude of the velocity (the speed) can
    change
  • The direction of the velocity can change
  • Even though the magnitude is constant
  • Both the magnitude and the direction can change

35
3.4 Projectile Motion
  • An object may move in both the x and y directions
    simultaneously
  • It moves in two dimensions
  • The form of two dimensional motion we will deal
    with is called projectile motion

36
Assumptions of Projectile Motion
  • We may ignore air friction
  • We may ignore the rotation of the earth
  • With these assumptions, an object in projectile
    motion will follow a parabolic path

37
Rules of Projectile Motion
  • The x- and y-directions of motion are completely
    independent of each other
  • The x-direction is uniform motion
  • ax 0
  • The y-direction is free fall
  • ay -g
  • The initial velocity can be broken down into its
    x- and y-components

38
Projectile Motion
39
Projectile Motion at Various Initial Angles
  • Complementary values of the initial angle result
    in the same range
  • The heights will be different
  • The maximum range occurs at a projection angle of
    45o

40
Some Details About the Rules
  • x-direction
  • ax 0
  • x vxot
  • This is the only operative equation in the
    x-direction since there is uniform velocity in
    that direction

41
More Details About the Rules
  • y-direction
  • free fall problem
  • a -g
  • take the positive direction as upward
  • uniformly accelerated motion, so the motion
    equations all hold

42
Velocity of the Projectile
  • The velocity of the projectile at any point of
    its motion is the vector sum of its x and y
    components at that point
  • Remember to be careful about the angles quadrant

43
Problem-Solving Strategy
  • Select a coordinate system and sketch the path of
    the projectile
  • Include initial and final positions, velocities,
    and accelerations
  • Resolve the initial velocity into x- and
    y-components
  • Treat the horizontal and vertical motions
    independently

44
Problem-Solving Strategy, cont
  • Follow the techniques for solving problems with
    constant velocity to analyze the horizontal
    motion of the projectile
  • Follow the techniques for solving problems with
    constant acceleration to analyze the vertical
    motion of the projectile

45
  • An object may be fired horizontally
  • The initial velocity is all in the x-direction
  • vo vx and vy 0
  • All the general rules of projectile motion apply

46
  • Follow the general rules for projectile motion
  • Break the y-direction into parts
  • up and down
  • symmetrical back to initial height and then the
    rest of the height
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