1. Should be: Wednesday, Sept 8

2. Announcements
You may attend any of the instructors’ office hours. List maintained on Blackboard under “Staff / Office Hours”.
HW #1 due tonight at 11:59 p.m.
HW #2 now available, due Thurs Sept 9 at 11:59 p.m.
Solutions to HW #1 will be available on Blackboard (under “Solutions”) after the due date/time.
Quiz grades will be posted to Blackboard after your recitation instructor has graded the quizzes (also returned next week).

3. Last Time: Motion in One Dimension
Displacement, Velocity, Acceleration
Today:
One-Dimensional Motion with Constant Acceleration
Freely Falling Objects (under gravity)

4. Constant/Uniform Acceleration
Constant/Uniform Acceleration: magnitude and direction of the acceleration does not change.
Constant acceleration is important, because it applies to many natural phenomena.
One such example is objects in “free fall” near the surface of the Earth. Neglecting air resistance, all objects (independent of mass) “fall” with the same downward acceleration !

5. Constant Acceleration
Key Point:
Under constant acceleration, the instantaneous acceleration at any point in a time interval is equal to the average acceleration over the entire time interval.
This Means:
velocity
slope = a v at
Let: ti = 0, vi = v0, tf = t, and vf = v v0 v0
time t
(for constant a only)

6. Constant Acceleration
velocity
velocity
a < 0
a > 0 v v0
time
time Δt
Average velocity in any time interval Δt is :
(for constant a only)
Just the average of the velocities at the start/finish of the time interval !

7. Constant Acceleration
Recall: The displacement Δx
and
average velocity v
Set:
If ti = 0 :
(for constant a only)
Using v = v0 + at :
(for constant a only)

8. Constant Acceleration
velocity
slope = a v
For constant acceleration, the displacement Δx is just equal to the area under the velocity-vs-time graph ! at v0 v0
time t
Recall: Area of Triangle = ½ (base) (height)

9. Constant Acceleration
Finally: Recall
and
Plugging in for t gives us:
(for constant a only)

10. “Equations of Motion” for Constant Acceleration
Information
Velocity as a function of time
Displacement as a function of time
Velocity as a function of displacement
These are for motion along a single axis (x-axis) with acceleration along that direction.
If acceleration and motion along y-axis, just change x  y.
Assumes that at t = 0, v = v0.
Recall: Δx = x – x0

11. Example
A car starting from rest undergoes a constant acceleration of a = 2.0 m/s2.
What is the velocity of the car after it has traveled 100 m?
How much time does it take to travel the 100 m?

12. Example: Problem 2.27
A car traveling east at 40.0 m/s (= mph !) passes a trooper hiding at roadside. The driver uniformly reduces his speed to 25.0 m/s in 3.50 s.
What is the magnitude and direction of the car’s acceleration as it slows down?
How far does the car travel during the 3.50-s time period?
v0 = 40 m/s
West
East -x +x

13. Example: Problem 2.29
A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final velocity of 2.80 m/s.
Find the truck’s original speed.
Find its acceleration.

14. Freely Falling Objects
Key Point:
When air resistance is negligible, all objects falling under the influence of gravity near the Earth’s surface fall at the same constant acceleration.
Aristotle (384 – 322 B.C.)
Plato (428 – 348 B.C.)
Heavier objects fall faster than lighter objects.

15. Galileo Galilei (1564-1642) Italian physicist and philosopher
Professor at the University of Padua
Invented the thermometer and the pendulum clock
“Galileo, perhaps more than any other single person, was responsible for the birth of modern science.”
– Stephen Hawking

16. Galileo Galilei ( )
First to observe the heavens with a telescope
Defended the (disturbing) idea of Copernicus ( ) that the Earth was not the center of the universe
Tried for heresy by Catholic Church (1992: Vatican found him not guilty)
Undertook series of experimental studies to describe the motion of bodies: mechanics

17. Freely Falling Objects
What is a freely falling object ?
Any object moving freely under the influence of gravity alone, regardless of its initial motion.
Denote magnitude of free-fall acceleration as:
g = 9.80 m/s2 (magnitude on the surface of the Earth)
If we:
Neglect air resistance
Assume acceleration doesn’t vary with altitude (over short vertical distances)
“up” = +y
Motion of freely falling object is the same as one-dimensional motion with constant acceleration !

18. “Equations of Motion” for Freely Falling Objects
+y = “up”
Equation
Information
Velocity as a function of time
Displacement as a function of time
Velocity as a function of displacement
Recall: Δy = y – y0

19. Objects Rising Against Gravity
If object’s initial (upward) velocity is v0 , how high will it rise? y
y = 0

20. Objects Rising Against Gravity
If object’s initial (upward) velocity is v0 , how high will it rise?
First, solve for the time to reach the highest point. y
 At highest point, v = 0 !
Second, calculate the maximum height after solving for the time to reach the highest point.
Alternative method …
y = 0

21. Example: Multiple Choice #10 (p. 48)
A student at the top of a building throws a red ball upward with speed v0, and then throws a green ball downwards with the same initial speed v0. Immediately before the balls hit the ground …
True or False: The velocities of the two balls are equal. v0 h y
y = 0
y = h

22. Example: Multiple Choice #10 (p. 48)
A student at the top of a building throws a red ball upward with speed v0, and then throws a green ball downwards with the same initial speed v0. Immediately before the balls hit the ground …
True or False: The speed of each ball is greater than v0. v0 h y
y = 0
y = h

23. Example: Multiple Choice #10 (p. 48)
A student at the top of a building throws a red ball upward with speed v0, and then throws a green ball downwards with the same initial speed v0. Immediately before the balls hit the ground …
True or False: The acceleration of the green ball is greater than that of the red ball. v0 h y
y = 0
y = h

24. Example: Problem 2.50
A small mailbag is released from a helicopter that is descending steadily at 1.50 m/s. After 2.00 s:
What is the speed of the mailbag? ; and
How far is it below the helicopter ?
How do (a) and (b) change if the helicopter is rising steadily at 1.50 m/s?

25. Reading Assignment
Next class: 3.1 – 3.3