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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
• 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
• 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
• 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
• 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

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