Kinematics and One Dimensional Motion: Non-Constant Acceleration 8.01 W01D3

1
Kinematics and One Dimensional
MotionNon-Constant Acceleration8.01W01D3
2
Todays Reading Assignment W02D1

• Young and Freedman Sections 2.1-2.6

3
Kinematics and One Dimensional Motion
4
Kinematics Vocabulary

• Kinema means movement in Greek
• Mathematical description of motion
• Position
• Time Interval
• Displacement
• Velocity absolute value speed
• Acceleration
• Averages of the latter two quantities.

5
Coordinate System in One Dimension

• Used to describe the position of a point in space
• A coordinate system consists of
• An origin at a particular point in space
• A set of coordinate axes with scales and labels
• Choice of positive direction for each axis unit
  vectors
• Choice of type Cartesian or Polar or Spherical
• Example Cartesian One-Dimensional Coordinate
  System

6
Position

• A vector that points from origin to body.
• Position is a function of time
• In one dimension

7
Displacement Vector

• Change in position vector of the object during
  the time interval

8
Concept Question Displacement
An object goes from one point in space to
another. After the object arrives at its
destination, the magnitude of its displacement
is 1) either greater than or equal to
2) always greater than 3) always equal to
4) either smaller than or equal to 5) always
smaller than 6) either smaller or larger than
the distance it traveled.
9
Average Velocity
The average velocity, , is the
displacement divided by the time
interval The x-component of the average
velocity is given by
10
Instantaneous Velocityand Differentiation

• For each time interval , calculate the
  x-component of the average velocity
• Take limit as sequence of the
  x-component average velocities
• The limiting value of this sequence is
  x-component of the instantaneous velocity at the
  time t .

11
Instantaneous Velocity
x-component of the velocity is equal to the slope
of the tangent line of the graph of x-component
of position vs. time at time t
12
Concept Question One Dimensional Kinematics

• The graph shows the position as a function of
  time for two trains running on parallel tracks.
  For times greater than t 0, which of the
  following is true
• At time tB, both trains have the same velocity.
• Both trains speed up all the time.
• Both trains have the same velocity at some time
  before tB, .
• Somewhere on the graph, both trains have the same
  acceleration.

13
Average Acceleration

• Change in instantaneous velocity divided by the
  time interval
• The x-component of the average acceleration

14
Instantaneous Accelerationand Differentiation

• For each time interval , calculate the
  x-component of the average acceleration
  
• Take limit as sequence of the
  x-component average accelerations
• The limiting value of this sequence is
  x-component of the instantaneous acceleration at
  the time t .

15
Instantaneous Acceleration
The x-component of acceleration is equal to the
slope of the tangent line of the graph of the
x-component of the velocity vs. time at time t
16
Table Problem Model Rocket

• A person launches a home-built model rocket
  straight up into the air at y 0 from rest at
  time t 0 . (The positive y-direction is
  upwards). The fuel burns out at t t0. The
  position of the rocket is given by
•  
•  
• with a0 and g are positive. Find the
  y-components of the velocity and acceleration of
  the rocket as a function of time. Graph ay vs
  t for 0 lt t lt t0.
•  
•  

17
Non-Constant Accelerationand Integration
18
Velocity as the Integral of Acceleration

• The area under the graph of the x-component of
  the acceleration vs. time is the change in
  velocity

19
Position as the Integral of Velocity

• Area under the graph of x-component of the
  velocity vs. time is the displacement

Ei is the error in the approximation for each
interval
20
Summary Time-Dependent Acceleration

• Acceleration is a non-constant function of time
• Change in velocity
• Change in position

21
Table Problem Sports Car

• At t 0 , a sports car starting at rest at x
  0 accelerates with an x-component of
  acceleration given by
• and zero afterwards with
• Find expressions for the velocity and position
  vectors of the sports as functions of time for t
  gt0.
• (2) Sketch graphs of the x-component of the
  position, velocity and acceleration of the sports
  car as a function of time for
• t gt0

22
Concept Question

• A particle, starting at rest at t 0,
  experiences a non-constant acceleration ax(t) .
  Its change of position can be found by
• Differentiating ax(t) twice.
• Integrating ax(t) twice.
• (1/2) ax(t) times t2.
• None of the above.
• Two of the above.

23
Next Reading Assignment

• Young and Freedman Sections 3.1-3.3, 3.5