1. Kinematics and One Dimensional Motion: Non-Constant Acceleration 8
Kinematics and One Dimensional Motion: Non-Constant Acceleration W01D3

2. Today’s Reading Assignment: W02D1
Young and Freedman: Sections

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.
Added time as a physical quantity used to describe motion

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 OK

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

7. Displacement Vector
Change in position vector of the object during the time interval OK

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
Clarify notation to students: the upper bar on v(t) or v_x(t) denotes the average. The arrow on the first bold v(t) denotes the vector nature.

10. Instantaneous Velocity and 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 .
Bullet list

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
OK, but graph is a little hard to read. Can we make the lines, and the text thicker next time?

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 OK

14. Instantaneous Acceleration and 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 .
Bullet list

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
Graph is a little hard to read. Hopefully it looks okay on the screen. Can we make the lines, and the text thicker next time?

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 < t < t0.

17. Non-Constant Acceleration and Integration17

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
Use (t-t_0) for time interval??

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 >0.
(2) Sketch graphs of the x-component of the position, velocity and acceleration of the sports car as a function of time for
t >0

22. Concept Question
A particle, starting at rest at t = 0, experiences a non-constant acceleration ax(t) . It’s 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 , 3.5