Motion in One Dimension

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Chapter 2

• Motion in One Dimension

Conceptual questions 3, 7, 10, 11, 15 Problems
11, 22, 59
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Dynamics

• The branch of physics involving the motion of an
  object and the relationship between that motion
  and other physics concepts
• Kinematics is a part of dynamics
• In kinematics, you are interested in the
  description of motion

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Vector and Scalar Quantities

• Vector quantities need both magnitude (size) and
  direction to completely describe them
• Represented by an arrow, the length of the arrow
  is proportional to the magnitude of the vector
• Head of the arrow represents the direction
• Generally printed in bold face type
• Scalar quantities are completely described by
  magnitude only

4
Displacement

• Measures the change in position
• Represented as ?x (if horizontal) or ?y (if
  vertical)
• Vector quantity
• or - is generally sufficient to indicate
  direction for one-dimensional motion
• Units are meters (m) in SI, centimeters (cm) in
  cgs or feet (ft) in US Customary

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Distance

• Distance may be, but is not necessarily, the
  magnitude of the displacement
• Blue line shows the distance
• Red line shows the displacement

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Velocity

• It takes time for an object to undergo a
  displacement
• The average velocity is rate at which the
  displacement occurs
• D represents the difference between the final and
  initial values
• Units of velocity are m/s (SI), cm/s (cgs) or
  ft/s (US Cust.)

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Speed

• Speed is a scalar quantity
• same units as velocity
• total distance / total time
• May be, but is not necessarily, the magnitude of
  the velocity

Instantaneous Velocity

• The limit of the average velocity as the time
  interval becomes infinitesimally short, or as the
  time interval approaches zero
• The instantaneous velocity indicates what is
  happening at every point of time

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Graphical Interpretation of Velocity

• Velocity can be determined from a position-time
  graph
• Average velocity equals the slope of the line
  joining the initial and final positions
• Instantaneous velocity is the slope of the
  tangent to the curve at the time of interest
• The instantaneous speed is the magnitude of the
  instantaneous velocity

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Average Velocity
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Instantaneous Velocity
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Problem

• Figure shows a position-time graph for several
  trains traveling along a straight track. Identify
  the graphs that correspond to (a) forward motion
  only, (b) backward motion only, (c) constant
  velocity, (d) greatest constant velocity, (e) no
  movement.

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Problem 2-11

• A person takes a trip driving with a constant
  speed of 89.5 km/h, except for a 22.0 min rest
  stop. If the persons average speed is 77.8
  km/h, how much time is spent on the trip and how
  far does the person travel?

13
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)

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Graphical Interpretation of Acceleration

• Average acceleration is the slope of the line
  connecting the initial and final velocities on a
  velocity-time graph
• Instantaneous acceleration is the slope of the
  tangent to the curve of the velocity-time graph

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Average Acceleration
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Problem 2-22
The velocity-time graph for an object moving
along a straight path is shown in Figure P2.22.
(a) Find the average accelerations of this
object during the time intervals 0 to 5.0 s, 5.0
s to 15 s, and 0 to 20 s. (b) Find the
instantaneous accelerations at 2.0 s, 10 s, and
18 s
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Relationship Between Acceleration and Velocity

• Uniform velocity (shown by red arrows maintaining
  the same size)
• Acceleration equals zero

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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)

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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)

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Conceptual questions
3. If a car is traveling eastward, can its
acceleration be westward? 10. Car A, traveling
from New York to Miami, has a speed of 25 m/s.
Car B, traveling from New York to Chicago, also
has a speed of 25 m/s. Are their velocities
equal? 11. A ball is thrown vertically upward.
What are its velocity and acceleration when it
reaches its maximum height? What is the
acceleration of the ball just before it hits the
ground?
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Kinematic Equations

• Motion with uniform acceleration

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Graphical Interpretation of the Equation
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Problem-Solving Hints

• Be sure all the units are consistent
• Convert if necessary
• Choose a coordinate system
• Sketch the situation, labeling initial and final
  points, indicating a positive direction
• Choose the appropriate kinematic equation
• Check your results

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Quick quiz 2.3

• Match each velocity-time graph with the
  acceleration-time graph that best describes the
  motion

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Free Fall

• All objects moving under the influence of only
  gravity are said to be in free fall
• 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
• Symbolized by g
• g 9.8 m/s²
• g is always directed downward, toward the center
  of the earth

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Free Fall -- an object dropped
y

• Initial velocity is zero, vo 0, acceleration a
  g
• Let up be positive
• Use the kinematic equations
• Generally use y instead of x since vertical

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Free Fall -- an object thrown downward

• a g
• Initial velocity ? 0
• With upward being positive, initial velocity will
  be negative

g
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Free Fall -- object thrown upward

• Initial velocity is upward, so positive
• The instantaneous velocity at the maximum height
  is zero
• a g everywhere in the motion
• g is always downward, negative

v 0
g
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Thrown upward, cont.

• The motion may be symmetrical
• then tup tdown
• then vf -vo
• The motion may be not symmetrical
• Break the motion into various parts
• generally up and down

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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

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Example 2.10
A rocket moves upward, starting from rest with an
acceleration a29.4 m/s2 for 4 s. It runs out of
fuel at the end of this 4 s and continues to move
upward for a while. How high does it rise above
its original starting point? What is its velocity
the instant before it crashes on the ground?
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Conceptual questions

• 7. A child throws a marble into the air with an
  initial speed vo. Another child drops a ball at
  the same instant. Compare the accelerations of
  the two objects while they are in flight.
• A ball is thrown vertically upward.
• What are its velocity and acceleration when it
  reaches its maximum altitude?
• (b) What is its acceleration just before it hits
  the ground?
• 15. A student at the top of a building throws one
  ball upward with the speed v and then throes
  another ball downward with the same initial speed
  v. How do the final velocities compare when the
  balls reach the ground?

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Problem 2-59
Two students are on a balcony 19.6 m above the
street. One student throws a ball vertically
downward at 14.7 m/s at the same instant the
other student throws a ball vertically upward at
the same speed. The second ball just misses the
balcony on the way down. (a) What is the
difference in their time in air? (b) What is the
velocity of each ball as it strikes the ground?
(c) How far apart are the balls 0.800 s after
they are thrown?