1. Chapter 4: Motion with a Changing Velocity
Motion Along a Line
Graphical Representation of Motion
Free Fall
Projectile Motion
Apparent Weight
Air Resistance and Terminal Velocity

2. §4.1 Motion Along a Line
For constant acceleration the kinematic equations are:
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 4, Questions 1 and 20.
Also:

3. Know: a= -2.94 m/s2, vix=10.0 m/s, vfx= 0.0 m/s Want: t.
Example: In a previous example, a box sliding across a rough surface was found to have an acceleration of m/s2. If the initial speed of the box is 10.0 m/s, how long does it take for the box to come to rest?
Know: a= m/s2, vix=10.0 m/s, vfx= 0.0 m/s
Want: t.
This example references the example from chapter 3 on slides 28 and 29.

4. Example (text problem 4.8): A train of mass 55,200 kg is traveling along a straight, level track at 26.8 m/s. Suddenly the engineer sees a truck stalled on the tracks 184 m ahead. If the maximum possible braking force has magnitude kN, can the train be stopped in time?
Know: vfx = 0 m/s, vix=26.8 m/s, x=184 m
Determine ax and compare to the train’s maximum ax.

5. The train’s maximum acceleration is:
Example continued:
The train’s maximum acceleration is:
The maximum acceleration is not sufficient to stop the train before it hits the stalled truck.

6. §4.2 Visualizing Motion with Constant Acceleration
Motion diagrams for three carts:
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 4, Questions 16 and 17.

7. Graphs of x, vx, ax for each of the three carts

8. Example (text problem 4.13): A trolley car in New Orleans starts from rest at the St. Charles Street stop and has a constant acceleration of 1.20 m/s2 for 12.0 seconds.
(a) Draw a graph of vx versus t.

9. (b) How far has the train traveled at the end of the 12.0 seconds?
Example continued:
(b) How far has the train traveled at the end of the 12.0 seconds?
The area between the curve and the time axis represents the distance traveled.
(c) What is the speed of the train at the end of the 12.0 s?
This can be read directly from the graph, vx=14.4 m/s.

10. §4.3 Free Fall
A stone is dropped from the edge of a cliff; if air resistance can be ignored, the FBD for the stone is: x y w
Apply Newton’s Second Law
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 4, Question 2.
The stone is in free fall; only the force of gravity acts on the stone.

11. Given: viy=+15.0 m/s; ay=-9.8 m/s2
Example: You throw a ball into the air with speed 15.0 m/s; how high does the ball rise? x y
viy ay
Given: viy=+15.0 m/s; ay=-9.8 m/s2
To calculate the final height, we need to know the time of flight.
Time of flight from:

12. The ball rises until vfy= 0.
Example continued:
The ball rises until vfy= 0.
The height:

13. Given: viy=0 m/s, ay=-9.8 m/s2, y=-369 m
Example (text problem 4.22): A penny is dropped from the observation deck of the Empire State Building 369 m above the ground. With what velocity does it strike the ground? Ignore air resistance. x y
Given: viy=0 m/s, ay=-9.8 m/s2, y=-369 m
369 m ay
Unknown: vyf
Use:

14. How long does it take for the penny to strike the ground?
Example continued:
(downward)
How long does it take for the penny to strike the ground?
Given: viy=0 m/s, ay=-9.8 m/s2, y=-369 m Unknown: t

15. §4.4 Projectile Motion
What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball.
The baseball has ax = 0 and ay = -g, it moves with constant velocity along the x-axis and with nonzero, constant acceleration along the y-axis.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 4, Questions 3, 4, 5, 6, 7, 12, 18, and 19.

16. Example: An object is projected from the origin
Example: An object is projected from the origin. The initial velocity components are vix = 7.07 m/s, and viy = 7.07 m/s. Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results.
Since the object starts from the origin, y and x will represent the location of the object at time t.

17. Example continued:
t (sec)
x (meters)
y (meters)
0.2
1.41
1.22
0.4
2.83
2.04
0.6
4.24
2.48
0.8
5.66
2.52
1.0
7.07
2.17
1.2
8.48
1.43
1.4
9.89
0.29

18. Example continued:
This is a plot of the x position (black points) and y position (red points) of the object as a function of time.

19. The object’s path is a parabola.
Example continued:
This is a plot of the y position versus x position for the object (its trajectory).
The object’s path is a parabola.

20. (a) What are vx and vy of the arrow when t=3 sec?
Example (text problem 4.36): An arrow is shot into the air with  = 60° and vi = 20.0 m/s.
(a) What are vx and vy of the arrow when t=3 sec? x y
60°
The components of the initial velocity are:
At t = 3 sec:

21. Example continued:
(b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval? y x r

22. Example: How far does the arrow in the previous example land from where it is released?
The arrow lands when y=0.
Solving for t:
The distance traveled is:

23. §4.5 Apparent Weight Stand on a bathroom scale. FBD: Nx y
FBD:
Apply Newton’s 2nd Law:
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 4, Questions 8, 9, 10 and 11.

24. The normal force is the force the scale exerts on you
The normal force is the force the scale exerts on you. By Newton’s 3rd Law this is also the force (magnitude only) you exert on the scale. A scale will read the normal force.
is what the scale reads.
When ay = 0, N = mg. The scale reads your true weight.
When ay0, N>mg or N<mg.

25. Example: A woman of mass 51 kg is standing in an elevator
Example: A woman of mass 51 kg is standing in an elevator. If the elevator pushes up on her feet with 408 newtons of force, what is the acceleration of the elevator? w N x y
FBD for woman:
Apply Newton’s 2nd Law:
(1)
This problem is number 89 in chapter 3. It was in this section in the first edition of the text but has been moved in the second edition.

26. Given: N = 408 newtons, m = 51 kg, g = 9.8 m/s2 Unknown: ay
Example continued:
Given: N = 408 newtons, m = 51 kg, g = 9.8 m/s2
Unknown: ay
Solving (1) for ay:
The elevator could be (1) traveling upward with decreasing speed, or (2) traveling downward with increasing speed.

27. §4.6 Air Resistance
A stone is dropped from the edge of a cliff; if air resistance cannot be ignored, the FBD for the stone is: x y w Fd
Apply Newton’s Second Law
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 4, Questions 13, 14, and 15.
Where Fd is the magnitude of the drag force on the stone. This force is directed opposite the object’s velocity

28. Assume that
b is a parameter that depends on the size and shape of the object.
Since Fdv2, can the object be in equilibrium?

29. Example (text problem 4.61): A paratrooper with a fully loaded pack has a mass of 120 kg. The force due to air resistance has a magnitude of Fd = bv2 where
b = 0.14 N s2/m2.
(a) If he/she falls with a speed of 64 m/s, what is the force of air resistance?

30. (b) What is the paratrooper’s acceleration?
Example continued:
(b) What is the paratrooper’s acceleration? x y w Fd
Apply Newton’s Second Law and solve for a.
FBD:
(c) What is the paratrooper’s terminal speed?

31. Summary The Kinematic Equations Graphical Representations of Motion
Applications of Newton’s Second Law & Kinematics (free fall, projectiles, accelerated motion, air drag)
Terminal Velocity