1. Kinematics II
April 20, 2017

2. Remember this graph?
WHAT ABOUT “B”??

3. Conclusion Velocity is not the same at all times. It is changing
A changing velocity is called an acceleration.

4. Acceleration in Pictures
Constant Velocity
Constant Acceleration

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

6. Example – Constant acceleration a=1 m/s2
Time (s)
Velocity (m/s) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7. Instantaneous Acceleration
The instantaneous acceleration is the limit of the average acceleration as t approaches 0

8. Calculus

9. Instantaneous Acceleration -- graph
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

10. Acceleration and Velocity, 1
When an object’s velocity and acceleration are in the same direction, the object is speeding up
When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

11. Example 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

12. Some math. Assume constant acceleration

13. More …

14. One more calculus trick: the chain rule

15. Kinematic Equations -- summary from the book.

16. 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

17. Kinematic Equations, specific
For constant a,
Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration
Does not give any information about displacement

18. Kinematic Equations, specific
For constant acceleration,
The average velocity can be expressed as the arithmetic mean of the initial and final velocities

19. Kinematic Equations, specific
For constant acceleration,
Gives final position in terms of velocity and acceleration
Doesn’t tell you about final velocity

20. 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

21. Graphical Look at Motion – velocity – time curve
The slope gives the acceleration
The straight line indicates a constant acceleration

22. Graphical Look at Motion – acceleration – time curve
The zero slope indicates a constant acceleration

23. This is called free fall.
Let’s get real and throw something out of a building and watch it fall.
This is called free fall.

24. 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

25. 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 Earth’s surface
In the English/American system, g=32 ft/sec2.

26. Acceleration of Free Fall/Rise, cont.
We will neglect air resistance
Free fall motion is constantly accelerated motion in one dimension
Use the kinematic equations to solve problems.

27. Free Fall Example Let’s drop an object from the top
of this building by simply releasing it.
How long will it take to get to the ground?
How fast will it be going when it gets to the
ground?
What color is the object?
Issues –
Where is the origin?
IS y the same thing as x?? How can that be???

28. Use Transparencies …..

29. Motion Equations from Calculus
Displacement equals the area under the velocity – time curve
The limit of the sum is a definite integral