1. Free Fall
Lecture 3

2. Reading Quiz: Throw ball up and let it return. Initial speed = v0
Reading Quiz: Throw ball up and let it return. Initial speed = v0. Round trip time is 2v0/g. What is the ball’s minimum speed during trip?
-v0 v0
-2v0
2v0

3. Reading Quiz: Throw ball up and let it return. Initial speed = v0
Reading Quiz: Throw ball up and let it return. Initial speed = v0. Round trip time is 2v0/g. What is minimum speed during trip?
-v0 v0
-2v0
2v0

4. Last Time Problem solving Distance, displacement
Speed; average, instantaneous velocity
Acceleration: average, instantaneous
Motion with constant acceleration

5. More problem solving strategy Free fall Acceleration of gravity
Today
More problem solving strategy
Free fall
Acceleration of gravity

6. Important equations for solving constant-acceleration problems.

7. Conceptual Quiz
If the average velocity is non-zero over some time interval, does this mean that the instantaneous velocity is never zero during the same interval?
A) yes
B) no
C) it depends
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8. Conceptual Quiz
If the average velocity is non-zero over some time interval, does this mean that the instantaneous velocity is never zero during the same interval?
A) yes
B) no
C) it depends
No!!! For example, your average velocity for a trip home might be 60 mph, but if you stopped for lunch on the way home, there was an interval when your instantaneous velocity was zero, in fact!

9. Solving Problems
Read the whole problem and make sure you understand it. Then read it again.
Decide on the objects under study and what the time interval is.
Draw a diagram and choose coordinate axes.
Write down the known (given) quantities, and then the unknown ones that you need to find.
What physics applies here? Plan an approach to a solution. Strategy!

10. Solving Problems, continued
6. Which equations relate the known and unknown quantities? Are they valid in this situation? Solve algebraically for the unknown quantities, and check that your result is sensible (correct dimensions). Solution.
7. Calculate the solution and round it to the appropriate number of significant figures, but do not round off for WebAssign answers.
8. Look at the result—is it reasonable? Does it agree with a rough estimate?
9. Check the units again.

11. Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity.
This is one of the most common examples of motion with constant acceleration.
Equal time intervals..
Figure Caption: Multiflash photograph of a falling apple, at equal time intervals. The apple falls farther during each successive interval, which means it is accelerating.

12. Freely falling objects
Most important example of constant acceleration; a = ± g = 9.81 m/s2
Do demo: Paper and racquetball
Nickel and feather
Galileo – father of physics
Galileo lived

13. We let x be downward. Look at one of our previous kinematic equations:
Let’s release an object at x0 = 0 at t = 0. We then also have v0 = 0, a = g. x
Note that g is always positive. Here x is down.

14. Our previous equations for v become
These are useful equations. Drop object from rest.

15. x1 = 4.9 m
Free fall from rest
x2 = 14.7 m
x3 = 24.5 m
x4 = 34.4 m

16. Freely Falling Objects
Acceleration due to gravity near Earth’s surface:
Direction of g: towards center of Earth y
Equations of motion for freely falling objects (using y instead of x:). Note that y is up in this case so a = -g.

17. So what would happen if we dropped a rope that had masses at equal intervals?
Do demo. Free fall
What would happen if we dropped a rope that had masses spaced out as t2?

18. What happens if we throw a ball up and let it fall back down?
We throw a ball up at x = 0 with speed v0.
What is its speed when it returns?
How long does it take to return?
How can we determine these numbers?
The equations we have determined must tell us these answers! Let’s use them.
x = 0

19. See problem solution video called Ball Up and Down
See problem solution video called Ball Up and Down. These videos are an important part of the course. You must go through them. I separate them as a way of breaking up the lecture.
This is Extra 1 Problem Solution Video. See my notebook lecture notes. Give them the time it takes the ball to go up and down.

20. What happens if we throw a ball up and let it fall back down?
We throw a ball up at x = 0 with speed v0.
What is its speed when it returns? v0
How long does it take to return? 2v0/g
x = 0

21. Conceptual Quiz
You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?
A) At the top of its trajectory
B) Its acceleration is constant everywhere
C) Halfway to the top of its trajectory
D) Just after it leaves your hand
E) Just before it returns to your hand on the way down
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22. Conceptual Quiz
You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?
A) At the top of its trajectory
B) Its acceleration is constant everywhere
C) Halfway to the top of its trajectory
D) Just after it leaves your hand
E) Just before it returns to your hand on the way down
The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g, which is constant at all points.

23. B) v0/g 2v0/g Can not be determined
Conceptual Quiz: Throw ball up and let it return. Initial speed = v0. Round trip time is 2v0/g. What is time when minimum speed is reached?
B) v0/g
2v0/g
Can not be determined

24. B) v0/g at top, so half way C) 2v0/g D) Can not be determined
Conceptual Quiz: Throw ball up and let it return. Initial speed = v0. Round trip time is 2v0/g. What is time when minimum speed is reached?
B) v0/g at top, so half way
C) 2v0/g
D) Can not be determined

25. Foot Race. Mary and Sally are in a foot race (see figure)
Foot Race. Mary and Sally are in a foot race (see figure). When Mary is 22 m from the finish line, she has a speed of 4.0 m/s and is 5.0 m behind Sally, who has a speed of 5.0 m/s. Sally thinks she has an easy win and so, during the remaining portion of the race, decelerates at a constant rate of 0.5 m/s2 to the finish line. What constant acceleration does Mary now need during the remaining portion of the race, if she wishes to cross the finish line side-by-side with Sally?
Giancoli, 4th ed, Problem 2-47.

26. Graphical Analysis and Numerical Integration
The total displacement of an object can be described as the area under the v-t curve:

27. Similarly, the velocity may be written as the area under the a-t curve.
However, if the velocity or acceleration is not integrable, or is known only graphically, numerical integration may be used instead.
Numerical integration is not as hard as it seems. It is easy to do with Excel.
See wolframalpha.com

28. Variable Acceleration; Integral Calculus
Deriving the kinematic equations through integration:
For constant acceleration,

29. Then:
For constant acceleration,

30. Conceptual Quiz A) both v = 0 and a = 0 B) v ¹ 0, but a = 0
When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path?
A) both v = 0 and a = 0
B) v ¹ 0, but a = 0
C) v = 0, but a ¹ 0
D) both v ¹ 0 and a ¹ 0
E) not really sure
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31. Conceptual Quiz A) both v = 0 and a = 0 B) v ¹ 0, but a = 0
When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path?
A) both v = 0 and a = 0
B) v ¹ 0, but a = 0
C) v = 0, but a ¹ 0
D) both v ¹ 0 and a ¹ 0
E) not really sure y
At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero! Otherwise it would remain at rest!!

32. Conceptual Quiz
You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you?
A) more than 10 m/s
B) 10 m/s
C) less than 10 m/s
D) zero
E) need more information
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33. Conceptual Quiz A) more than 10 m/s B) 10 m/s C) less than 10 m/s
You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you?
A) more than 10 m/s
B) 10 m/s
C) less than 10 m/s
D) zero
E) need more information
The ball is slowing down on the way up due to gravity. Eventually it stops. Then it accelerates downward due to gravity (again). Because a = g on the way up and on the way down, the ball reaches the same speed when it gets back to you as it had when it left.

34. Conceptual Quiz v0 Bill Alice vA vB v0 Bill Alice vA vB
Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true?
A) vA < vB
B) vA = vB
C) vA > vB
D) impossible to tell v0
Bill
Alice H vA vB v0
Bill
Alice H vA vB
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35. Conceptual Quiz v0 Bill Alice vA vB
Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true?
A) vA < vB
B) vA = vB
C) vA > vB
R) impossible to tell v0
Bill
Alice H vA vB
Bill’s ball goes up and comes back down to Bill’s level. At that point, it is moving downward with v0, the same as Alice’s ball. Thus, it will hit the ground with the same speed as Alice’s ball.