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Kinematics in one dimension

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We will consider objects to be point particles for simplicity (think ... moment during the trip, a speedometer, graphical analysis, or calculus must be used. ... – PowerPoint PPT presentation

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Title: Kinematics in one dimension


1
Chapter 2
  • Kinematics in one dimension

2
Kinematics
  • Description of how objects move
  • One dimensional translational motion is an object
    moving along a straight line path
  • We will consider objects to be point particles
    for simplicity (think about a runner

3
In Other Words
  • A complex look at something moving in a straight
    line.
  • This complexity is necessary for later use in the
    study of velocity and acceleration
  • You will eventually be able to solve the
    velocity, time, or position in the x and y
    directions at any point along this path.

4
Frame of Reference
5
Reference Frames and Displacement
We make a distinction between distance and
displacement. Displacement (blue line) is how
far the object is from its starting point,
regardless of how it got there. Distance traveled
(dashed line) is measured along the actual path.
6
Distance vs. Displacement
  • Mary Prakel starts at the finish line, makes 8
    grueling laps around the track, and stops exactly
    where she started.
  • What is her distance traveled?
  • What is her displacement?

7
Coordinate axes
  • Used to show direction
  • Position of an object at any moment is given by
    its x coordinate
  • Directions of the axes can be placed at our
    convenience.
  • Vertical motion often represented by the y axis

y
x
-x
-y
8
Vectors
  • Quantities that have both magnitude and
    direction.
  • Ex displacement, velocity, acceleration
  • Quantities such as distance and speed, which
    describe only magnitude, are called scalars.

9
Vectors continued
  • Vectors which point in one direction get a
    positive sign
  • Those in the opposite direction get a negative
    sign.
  • Just because a displacement vector is negative
    does not mean it is found in the x side of the
    line. (what would be an example of this?)

y
x
x2
x1
x
10
Vectors continued
  • Assume each division stands for 5 m
  • Displacement is solved using the formula ?x x2
    x1
  • ?x 20m 5m
  • Displacement is 15 m

y
x
x2
x1
x
11
Vectors continued
  • This vector shows a displacement of the same
    magnitude in the opposite direction.
  • ?x x2 x1 in this case is
  • ?x 5m 20m -15m
  • Displacement is -15m

y
x2
x1
x
12
Vectors continued
  • In these cases a vector pointing right is
    positive while a vector pointing left is
    negative.

y
x
13
Understanding Quantities
14
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15
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16
Speed vs. Velocity
  • Speed refers to how far an object travels in a
    certain amount of time regardless of direction.
  • Average speed distance traveled
  • time elapsed
  • Speed is always a positive number

17
Average Velocity
  • Velocity refers to an objects displacement in a
    certain amount of time.
  • Rate of change of position
  • Average velocity displacement
  • time elapsed
  • final position initial position
  • time elapsed
  • Can be positive or negative

18
Cruise control automatically keeps your car at a
constant speed. You head east for 4000 s with
the cruise control set at 31 m/s. You end up
150,000 m east of Versailles.
  • How far east of Versailles did you start out?
  • Sketch a graph of x vs. t for this motion.
  • How fast are you traveling in mph?

19
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20
about 70 mi/hr
21
Some Av. Velocity Examples
22
Solve the Average Velocity for A,B,C,D
23
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24
Now draw x vs. t graphs for the four cars.
25
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26
Check your understanding
  • Which of the cars have positive displacements?
  • Which graph has the steepest slope?
  • Put the cars speeds in order from greatest to
    least.
  • Which has a greater velocity B or D?
  • A and C (B and D have negative displacements)
  • B (C has the most gradual slope)
  • B, D, A, C.
  • D has the greater velocity. A more negative
    number is not greater than a less negative number.

27
Nonuniform Motion
  • Description of an object that does not maintain
    the same motion during the entire time interval
    being studied
  • In other words, it has changing velocity during
    the time being studied
  • The previous graphs represented uniform motion.
    Lets look at some nonuniform motion.

28
Nonuniform motion
  • Attempt to solve the average velocity for each of
    these vehicles between 0 and 4 s.
  • Then attempt to sketch a graph for each cars
    nonuniform motion. (not all graphs will be
    straight lines)

29
Average velocity
  • A is 50 m/s
  • B is 50 m/s
  • C is 50 m/s
  • D is -50 m/s
  • E is -50 m/s

30
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31
Interpreting slope for nonuniform motion
  • If you draw a straight line segment (secant)
    connecting points P1 and P2, its slope is the
    average velocity of the object between those two
    points
  • Which segment has the greater average velocity?

32
Average vs. Instantaneous Velocity
  • The average velocities we have been solving do
    not describe variations in the objects velocity
    throughout the trip.
  • The graphs just made do not necessarily represent
    the actual velocities of the cars at any instant
    of time between 0 and 4 s.
  • To find the velocity of an object at any moment
    during the trip, a speedometer, graphical
    analysis, or calculus must be used.

33
2-3 Instantaneous Velocity
The instantaneous velocity is the average
velocity, in the limit as the time interval
becomes infinitesimally short.
(2-3)
These graphs show (a) constant velocity and (b)
varying velocity.
34
Graphical analysis
  • To graphically solve the instantaneous velocity
    at any point on the curve, we must draw a tangent
    line that just grazes the curve and then solve
    its slope.
  • All curves are essentially made up of many
    straight lines.

35
Graphs of v vs. t
  • Graph a shows non uniform motion x vs. t.
  • The lines of tangent for P1-P3 show the
    instantaneous velocity at those points.
  • b shows the v vs. t graph for the same motion.

36
Notice
  • P1 slope is 0, so velocity is zero
  • P2 has the steepest slope, so it represents the
    greatest velocity.
  • P3 has a slope less than P2 so the velocity is
    less at P3 than at P2
  • This is the graphical representation of a
    derivative in calculus.
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