Motion in One Dimension

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Chapter 2

• Motion in One Dimension

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• 2.1 position, Velocity, and Speed
• 2.2 Instantaneous Velocity and Speed
• 2.3 Acceleration
• 2.4 Freely Falling Objects
• 2.5 Kinematic Equations Derived from Calculus.

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Kinematics

• Kinematics describes motion while ignoring the
  agents that caused the motion
• For now, we will consider motion in one dimension
• Along a straight line
• We will use the particle model
• A particle is a point-like object, that has mass
  but infinitesimal size

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Position

• Position is defined in terms of a frame of
  reference
• For one dimension the motion is generally along
  the x- or y-axis
• The objects position is its location with
  respect to the frame of reference

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Position-Time Graph

• The position-time graph shows the motion of the
  particle (car)
• The smooth curve is a guess as to what happened
  between the data points

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Displacement

• Displacement is defined as the change in position
  during some time interval
• Represented as ?x
• ?x xf - xi
• SI units are meters (m), ?x can be positive or
  negative
• Displacement is different than distance. Distance
  is the length of a path followed by a particle.

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Vectors and Scalars

• Vector quantities that need both -magnitude (size
  or numerical value) and direction to completely
  describe them
• We will use and signs to indicate vector
  directions
• Scalar quantities are completely described by
  magnitude only

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Average Velocity

• The average velocity is the rate at which the
  displacement occurs
• The dimensions are length / time L/T
• The SI units are m/s
• Is also the slope of the line in the position
  time graph

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Average Speed

• Speed is a scalar quantity
• same units as velocity
• total distance / total time
• The average speed is not (necessarily) the
  magnitude of the average velocity

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Instantaneous Velocity

• Instantaneous velocity is 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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Instantaneous Velocity

• The general equation for instantaneous velocity
  is
• The instantaneous velocity can be positive,
  negative, or zero

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Instantaneous Velocity

• The instantaneous velocity is the slope of the
  line tangent to the x vs t curve
• This would be the green line
• The blue lines show that as ?t gets smaller, they
  approach the green line

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Instantaneous Speed

• The instantaneous speed is the magnitude of the
  instantaneous velocity
• Remember that the average speed is not the
  magnitude of the average velocity

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Average Acceleration

• Acceleration is the rate of change of the
  velocity
• Dimensions are L/T2
• SI units are m/s²

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Instantaneous Acceleration

• The instantaneous acceleration is the limit of
  the average acceleration as ?t approaches 0

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Instantaneous Acceleration

• The slope of the velocity vs. time graph is the
  acceleration
• The green line represents the instantaneous
  acceleration
• The blue line is the average acceleration

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Acceleration and Velocity

• When an objects velocity and acceleration are in
  the same direction, the object is speeding up
• When an objects velocity and acceleration are in
  the opposite direction, the object is slowing
  down

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Acceleration and Velocity

• The car is moving with constant positive velocity
  (shown by red arrows maintaining the same size)
• Acceleration equals zero

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Acceleration and Velocity

• 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)
• This shows positive acceleration and positive
  velocity

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Acceleration and Velocity

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

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1D motion with constant acceleration
tf ti t
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1D motion with constant acceleration

• In a similar manner we can rewrite equation for
  average velocity
• and than solve it for xf
• Rearranging, and assuming

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1D motion with constant acceleration
(1)

• Using
• and than substituting into equation for final
  position yields

(1)
(2)
(2)
Equations (1) and (2) are the basic kinematics
equations
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1D motion with constant acceleration

• These two equations can be combined to yield
  additional equations.
• We can eliminate t to obtain
• Second, we can eliminate the acceleration a to
  produce an equation in which acceleration does
  not appear

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Kinematics with constant acceleration - Summary
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Kinematic Equations - summary
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Kinematic Equations

• The kinematic equations may be used to solve any
  problem involving one-dimensional motion with a
  constant acceleration
• You may need to use two of the equations to solve
  one problem
• Many times there is more than one way to solve a
  problem

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Kinematics - Example 1

• How long does it take for a train to come to rest
  if it decelerates at 2.0m/s2 from an initial
  velocity of 60 km/h?

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Kinematics - Example 1

• How long does it take for a train to come to rest
  if it decelerates at 2.0m/s2 from an initial
  velocity of 60 km/h?
• Using we rearrange to solve for t
• Vf 0.0 km/h, vi60 km/h and a -2.0 m/s2.

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• A car is approaching a hill at 30.0 m/s when its
  engine suddenly fails just at the bottom of the
  hill. The car moves with a constant acceleration
  of 2.00 m/s2 while coasting up the hill. (a)
  Write equations for the position along the slope
  and for the velocity as functions of time, taking
  x 0 at the bottom of the hill, where vi
  30.0 m/s. (b) Determine the maximum distance
  the car rolls up the hill.

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• (a) Take at the bottom of the
  hill where xi0, vi30m/s, a-2m/s2. Use these
  values in the general equation

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• (a) Take at the bottom of the
  hill where xi0, vi30m/s, a-2m/s2. Use these
  values in the general equation

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The distance of travel, xf,
becomes a maximum,
when
(turning point in the motion). Use the
expressions found in part (a) for
when t15sec.
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Graphical Look at Motion displacement-time curve

• The slope of the curve is the velocity
• The curved line indicates the velocity is
  changing
• Therefore, there is an acceleration

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Graphical Look at Motion velocity-time
curve

• The slope gives the acceleration
• The straight line indicates a constant
  acceleration

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Graphical Look at Motion acceleration-time curve

• The zero slope indicates a constant acceleration

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Freely Falling Objects

• A freely falling object is any object moving
  freely under the influence of gravity alone.
• It does not depend upon the initial motion of the
  object
• Dropped released from rest
• Thrown downward
• Thrown upward

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Acceleration of Freely Falling Object

• The acceleration of an object in free fall is
  directed downward, regardless of the initial
  motion
• The magnitude of free fall acceleration is
  g 9.80 m/s2
• g decreases with increasing altitude
• g varies with latitude
• 9.80 m/s2 is the average at the Earths surface

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Acceleration of Free Fall

• We will neglect air resistance
• Free fall motion is constantly accelerated motion
  in one dimension
• Let upward be positive
• Use the kinematic equations with
  ay g -9.80 m/s2

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Free Fall Example

• Initial velocity at A is upward () and
  acceleration is g (-9.8 m/s2)
• At B, the velocity is 0 and the acceleration is g
  (-9.8 m/s2)
• At C, the velocity has the same magnitude as at
  A, but is in the opposite direction

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• A student throws a set of keys vertically upward
  to her sorority sister, who is in a window 4.00 m
  above. The keys are caught 1.50 s later by the
  sister's outstretched hand. (a) With what initial
  velocity were the keys thrown? (b) What was the
  velocity of the keys just before they were caught?

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• A student throws a set of keys vertically upward
  to her sorority sister, who is in a window 4.00 m
  above. The keys are caught 1.50 s later by the
  sister's outstretched hand. (a) With what initial
  velocity were the keys thrown? (b) What was the
  velocity of the keys just before they were caught?

(a)
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• A student throws a set of keys vertically upward
  to her sorority sister, who is in a window 4.00 m
  above. The keys are caught 1.50 s later by the
  sister's outstretched hand. (a) With what initial
  velocity were the keys thrown? (b) What was the
  velocity of the keys just before they were caught?

(b)
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• A ball is dropped from rest from a height h
  above the ground. Another ball is thrown
  vertically upwards from the ground at the instant
  the first ball is released. Determine the speed
  of the second ball if the two balls are to meet
  at a height h/2 above the ground.

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• A ball is dropped from rest from a height h
  above the ground. Another ball is thrown
  vertically upwards from the ground at the instant
  the first ball is released. Determine the speed
  of the second ball if the two balls are to meet
  at a height h/2 above the ground.

1-st ball
2-nd ball
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• A freely falling object requires 1.50 s to
  travel the last 30.0 m before it hits the ground.
  From what height above the ground did it fall?

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• A freely falling object requires 1.50 s to
  travel the last 30.0 m before it hits the ground.
  From what height above the ground did it fall?

Consider the last 30 m of fall. We find its speed
30 m above the ground
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• A freely falling object requires 1.50 s to
  travel the last 30.0 m before it hits the ground.
  From what height above the ground did it fall?

Now consider the portion of its fall above the 30
m point. We assume it starts from rest
Its original height was then
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Motion Equations from Calculus

• Displacement equals the area under the velocity
  time curve
• The limit of the sum is a definite integral

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Kinematic Equations General Calculus Form
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Kinematic Equations Calculus Form with Constant
Acceleration

• The integration form of vf vi gives
• The integration form of xf xi gives

52

• The height of a helicopter above the ground is
  given by h 3.00t3, where h is in meters
  and t is in seconds. After 2.00 s, the
  helicopter releases a small mailbag. How long
  after its release does the mailbag reach the
  ground?

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• The height of a helicopter above the ground is
  given by h 3.00t3, where h is in meters
  and t is in seconds. After 2.00 s, the
  helicopter releases a small mailbag. How long
  after its release does the mailbag reach the
  ground?

54

• The height of a helicopter above the ground is
  given by h 3.00t3, where h is in meters
  and t is in seconds. After 2.00 s, the
  helicopter releases a small mailbag. How long
  after its release does the mailbag reach the
  ground?

The equation of motion of the mailbag is
55

• Automotive engineers refer to the time rate of
  change of acceleration as the "jerk." If an
  object moves in one dimension such that its jerk
  J is constant, (a) determine expressions for its
  acceleration ax(t), velocity vx(t), and position
  x(t), given that its initial acceleration, speed,
  and position are axi , vxi, and xi ,
  respectively. (b) Show that

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• Automotive engineers refer to the time rate of
  change of acceleration as the "jerk." If an
  object moves in one dimension such that its jerk
  J is constant, (a) determine expressions for its
  acceleration ax(t), velocity vx(t), and position
  x(t), given that its initial acceleration, speed,
  and position are axi , vxi, and xi ,
  respectively. (b) Show that

(a)
constant
when
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• Automotive engineers refer to the time rate of
  change of acceleration as the "jerk." If an
  object moves in one dimension such that its jerk
  J is constant, (a) determine expressions for its
  acceleration ax(t), velocity vx(t), and position
  x(t), given that its initial acceleration, speed,
  and position are axi , vxi, and xi ,
  respectively. (b) Show that

(a)
when
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• Automotive engineers refer to the time rate of
  change of acceleration as the "jerk." If an
  object moves in one dimension such that its jerk
  J is constant, (a) determine expressions for its
  acceleration ax(t), velocity vx(t), and position
  x(t), given that its initial acceleration, speed,
  and position are axi , vxi, and xi ,
  respectively. (b) Show that

(a)
when
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(b)
Recall the expression for v
60

• The acceleration of a marble in a certain fluid
  is proportional to the speed of the marble
  squared, and is given (in SI units) by a 3.00
  v2 for v gt 0. If the marble enters this fluid
  with a speed of 1.50 m/s, how long will it take
  before the marble's speed is reduced to half of
  its initial value?

61

• The acceleration of a marble in a certain fluid
  is proportional to the speed of the marble
  squared, and is given (in SI units) by a 3.00
  v2 for v gt 0. If the marble enters this fluid
  with a speed of 1.50 m/s, how long will it take
  before the marble's speed is reduced to half of
  its initial value?

62

• The acceleration of a marble in a certain fluid
  is proportional to the speed of the marble
  squared, and is given (in SI units) by a 3.00
  v2 for v gt 0. If the marble enters this fluid
  with a speed of 1.50 m/s, how long will it take
  before the marble's speed is reduced to half of
  its initial value?

63

• A test rocket is fired vertically upward from a
  well. A catapult gives it initial velocity 80.0
  m/s at ground level. Its engines then fire and
  it accelerates upward at 4.00 m/s2 until it
  reaches an altitude of 1 000 m. At that point
  its engines fail and the rocket goes into free
  fall, with an acceleration of 9.80 m/s2. (a) How
  long is the rocket in motion above the ground?
  (b) What is its maximum altitude? (c) What is its
  velocity just before it collides with the Earth?

64

• A test rocket is fired vertically upward from a
  well. A catapult gives it initial velocity 80.0
  m/s at ground level. Its engines then fire and
  it accelerates upward at 4.00 m/s2 until it
  reaches an altitude of 1 000 m. At that point
  its engines fail and the rocket goes into free
  fall, with an acceleration of 9.80 m/s2. (a) How
  long is the rocket in motion above the ground?
  (b) What is its maximum altitude? (c) What is its
  velocity just before it collides with the Earth?

Let point 0 be at ground level and point 1 be at
the end of the engine burn. Let point 2 be the
highest point the rocket reaches and point 3 be
just before impact. The data in the table are
found for each phase of the rockets motion.
(0 to 1)
so

65

• A test rocket is fired vertically upward from a
  well. A catapult gives it initial velocity 80.0
  m/s at ground level. Its engines then fire and
  it accelerates upward at 4.00 m/s2 until it
  reaches an altitude of 1 000 m. At that point
  its engines fail and the rocket goes into free
  fall, with an acceleration of 9.80 m/s2. (a) How
  long is the rocket in motion above the ground?
  (b) What is its maximum altitude? (c) What is its
  velocity just before it collides with the Earth?

(1 to 2)
This is the time of maximum height of the rocket.
66

• A test rocket is fired vertically upward from a
  well. A catapult gives it initial velocity 80.0
  m/s at ground level. Its engines then fire and
  it accelerates upward at 4.00 m/s2 until it
  reaches an altitude of 1 000 m. At that point
  its engines fail and the rocket goes into free
  fall, with an acceleration of 9.80 m/s2. (a) How
  long is the rocket in motion above the ground?
  (b) What is its maximum altitude? (c) What is its
  velocity just before it collides with the Earth?

(2 to 3)
(a)
(b)
(c)
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t x v a
0 Launch 0.0 0 80 4.00
1 End Thrust 10.0 1 000 120 4.00
2 Rise Upwards 22.2 1 735 0 9.80
3 Fall to Earth 41.0 0 184 9.80
68

• An inquisitive physics student and mountain
  climber climbs a 50.0-m cliff that overhangs a
  calm pool of water. He throws two stones
  vertically downward, 1.00 s apart, and observes
  that they cause a single splash. The first stone
  has an initial speed of 2.00 m/s. (a) How long
  after release of the first stone do the two
  stones hit the water? (b) What initial velocity
  must the second stone have if they are to hit
  simultaneously? (c) What is the speed of each
  stone at the instant the two hit the water?

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(a) Only the positive root is
physically meaningful after the first stone
is thrown.

• An inquisitive physics student and mountain
  climber climbs a 50.0-m cliff that overhangs a
  calm pool of water. He throws two stones
  vertically downward, 1.00 s apart, and observes
  that they cause a single splash. The first stone
  has an initial speed of 2.00 m/s. (a) How long
  after release of the first stone do the two
  stones hit the water? (b) What initial velocity
  must the second stone have if they are to hit
  simultaneously? (c) What is the speed of each
  stone at the instant the two hit the water?

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(b)
downward
(c)
downward
downward