Motion in One Dimension

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Chapter 2

• Motion in One Dimension

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Dynamics

• The branch of physics involving the motion of an
  object and the relationship between that motion
  to the objects mass and forces on that object
• Kinematics is a part of dynamics
• In kinematics, you are interested in the
  description of motion
• Not concerned with the cause of the motion
  (forces causing the motion)

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Brief History of Motion

• Sumaria and Egypt
• Mainly motion of heavenly bodies
• Greeks
• Also to understand the motion of heavenly bodies
• Systematic and detailed studies

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Modern Ideas of Motion

• Galileo
• Made astronomical observations with a telescope
• Experimental evidence for description of motion
• Quantitative study of motion
• Motion of objects rolling down inclined planes

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Position

• Defined in terms of a frame of reference
• One dimensional, so generally the x- or y-axis
• DISPLACEMENT
• Dxxf - xi

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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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Displacements
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Scalar Quantities

• Scalar quantities are completely described by
  magnitude only
• Temperature is an example of a scalar
• quantity also distance and speed.

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Vector Quantities

• Vector quantities need both magnitude (size) and
  direction to completely describe them (scalar
  direction)
• 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

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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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Problem

• A guy decides to get some exercise and run to the
  end
• of his block a distance d. He decides that hes
  had
• enough exercise and runs back home.
• What was his displacement?
• What was the total distance he ran?

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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
• generally use a time interval, so let ti 0

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Velocity continued

• Direction will be the same as the direction of
  the displacement (time interval is always
  positive)
• or - is sufficient
• Units of velocity are m/s (SI), cm/s (cgs) or
  ft/s (US Cust.)
• Other units may be given in a problem, but
  generally will need to be converted to these

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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

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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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Uniform Velocity

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

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Problem
See problem 3 (page 50)
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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
See problem 7 (page 50)
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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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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

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Instantaneous and Uniform Acceleration

• The limit of the average acceleration as the time
  interval goes to zero
• When the instantaneous accelerations are always
  the same, the acceleration will be uniform
• The instantaneous accelerations will all be equal
  to the average acceleration

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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
See problem 18 (page 51)
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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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Problem
See problem 22 (page 51)
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Kinematic Equations

• Used in situations with uniform acceleration

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Notes on the equations

• Gives displacement as a function of velocity and
  time

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Notes on the equations

• Shows velocity as a function of acceleration and
  time

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Graphical Interpretation of the Equation
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Notes on the equations

• Gives displacement as a function of time,
  velocity and acceleration

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Notes on the equations

• Gives velocity as a function of acceleration and
  displacement

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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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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
• Galileo originated our present ideas about free
  fall from his inclined planes
• The acceleration is called the acceleration due
  to gravity, and indicated by g

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Acceleration due to Gravity

• 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

• Initial velocity is zero
• Let up be positive
• Use the kinematic equations
• Generally use y instead of x since vertical

vo 0 a g
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Problem

• A guy at the top of the empire state building
• decides to drop a penny off the top just to see
• if it really will get stuck in the sidewalk 381
• meters below.
• What is the velocity of the penny when it
• hits the sidewalk below?
• (b) How long does it take for the penny to reach
  the ground?

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Free Fall -- an object thrown downward

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

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Problem

• The same guy at the top of the empire state
• building decides to throw the a penny off the
• top with and initial velocity of 30m/s.
• What is the velocity of the penny when it
• hits the sidewalk below?
• (b) How long does it take this time for the penny
  to reach the ground?

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Free Fall -- object thrown upward
v 0 m/s

• 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

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Problem

• Roger Clemens throws a baseball straight up
• into the air with an initial velocity of of 50
  m/s.
• How high does it go?
• (b) How long does it take to get to its maximum
  height?

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Thrown upward, cont.

• The motion may be symmetrical
• then tup tdown
• then vf -vo
• The motion may not be 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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Combination Motions
Example 2.9 Page 46