1. College Physics, 7th Edition
Lecture Outline
Chapter 3
College Physics, 7th Edition
Wilson / Buffa / Lou
© 2010 Pearson Education, Inc.

2. Chapter 3 Motion in Two Dimensions
© 2010 Pearson Education, Inc.

3. Units of Chapter 3 Components of Motion
Vector Addition and Subtraction
Projectile Motion
Relative Velocity
© 2010 Pearson Education, Inc.

4. 3.1 Components of Motion
Motion in a straight line indicates only 1 axis needed.
What if motion is not in a straight line?
What do we do?

5. 3.1 Components of Motion
An object in motion on a plane can be located using two numbers—the x and y coordinates of its position. Similarly, its velocity can be described using components along the x- and y-axes.
© 2010 Pearson Education, Inc.

6. 3.1 Components of Motion The velocity components are:
Why does the x component use cos? y use sin?
The magnitude of the velocity vector is:
© 2010 Pearson Education, Inc.

7. 3.1 Components of Motion
The components of the displacement are then given by:
*Note that the x- and y-components are calculated separately.
© 2010 Pearson Education, Inc.

8. 3.1 Components of Motion Example #1:
If a diagonally moving ball has a constant velocity of 0.50 m/s at an angle of 37 degrees relative to the x axis, find how far it travels in 3.0s by using x and y components of its motion.

9. 3.1 Components of Motion The equations of motion are:
What do these equations have in common with our kinematic equations?
When solving two-dimensional kinematics problems, each component is treated separately. The time is common to both.
© 2010 Pearson Education, Inc.

10. 3.1 Components of Motion
If an object is initially moving with constant velocity and experiences acceleration in the direction of or opposite….
If the acceleration is not parallel to the velocity, the object will move in a curve: (aka…)
Figure 3.2
© 2010 Pearson Education, Inc.

11. 3.1 Components of Motion Example #1:
Suppose that the ball in Figure 3.2 has an initial velocity of 1.50 m/s along the x axis. Starting at t = 0, the ball receives an acceleration of 2.80 m/s2 in the y direction.
A.) What is the position of the ball 3.00s after t=0?
B.) What is the resultant velocity of the ball at that time?

12. 3.2 Vector Addition and Subtraction
Vector Addition is……..
Also called Vector Sum
Product is called the Resultant
You’ve been doing this already, you just don’t know it….when we add/subtract displacements

13. 3.2 Vector Addition and Subtraction
Geometric methods of vector addition
Triangle method: [We learned it last year as…]
© 2010 Pearson Education, Inc.

14. 3.2 Vector Addition and Subtraction
The negative of a vector has the same magnitude but is opposite in direction to the original vector. Adding a negative vector is the same as subtracting a vector.
© 2010 Pearson Education, Inc.

15. Question 3.1a Vectors I a) same magnitude, but can be in any direction
b) same magnitude, but must be in the same direction
c) different magnitudes, but must be in the same direction
d) same magnitude, but must be in opposite directions
e) different magnitudes, but must be in opposite directions
If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B?
Answer: d

16. 3.2 Vector Addition and Subtraction
Vector Components and the Analytical Component Method
If you know A and B, here is how to find C:
Magnitude Angle Form
© 2010 Pearson Education, Inc.

17. 3.2 Vector Addition and Subtraction
The components of C are given by:
Equivalently,
© 2010 Pearson Education, Inc.

18. Question 3.2a Vector Components I
a) it doubles
b) it increases, but by less than double
c) it does not change
d) it is reduced by half
e) it decreases, but not as much as half
If each component of a vector is doubled, what happens to the angle of that vector?
Answer: c

19. Question 3.3 Vector Addition
You are adding vectors of length 20 and 40 units. What is the only possible resultant magnitude that you can obtain out of the following choices?
a) 0
b) 18
c) 37
d) 64
e) 100
Answer: c

20. 3.2 Vector Addition and Subtraction
Vectors can also be written using unit vectors:
© 2010 Pearson Education, Inc.

21. 3.2 Vector Addition and Subtraction
Vectors can be resolved into components and the components added separately; then recombine to find the resultant.
© 2010 Pearson Education, Inc.

22. 3.2 Vector Addition and Subtraction

23. 3.2 Vector Addition and Subtraction
This is done most easily if all vectors start at the origin.
© 2010 Pearson Education, Inc.

24. 3.2 Vector Addition and Subtraction
Apply the component method to the addition of vectors for the Figure in the previous slide.
Finally, determine the magnitude of the resultant velocity as well as the angle.

25. 3.3 Projectile Motion
Projectile motion is 2-D motion of an object that is thrown or projected by some means.
When an object is projected, it is in free fall.
Suppose you throw an object horizontally with an initial velocity.
Is there a vertical acceleration?
Is there a horizontal acceleration?
What if no gravity??

26. Question 3.4a Firing Balls I
A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?
a) it depends on how fast the cart is moving
b) it falls behind the cart
c) it falls in front of the cart
d) it falls right back into the cart
e) it remains at rest
Answer: d

27. Question 3.4b Firing Balls II
Now the cart is being pulled along a horizontal track by an external force (a weight hanging over the table edge) and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball?
a) it depends upon how much the track is tilted
b) it falls behind the cart
c) it falls in front of the cart
d) it falls right back into the cart
e) it remains at rest
Answer: b

28. 3.3 Projectile Motion
One object dropped vertically (red) and another object projected horizontally (yellow).
What do you observe from the picture?

29. 3.3 Projectile Motion
An object projected horizontally has an initial velocity in the horizontal direction, and acceleration (due to gravity) in the vertical direction. The time it takes to reach the ground is the same as if it were simply dropped.
© 2010 Pearson Education, Inc.

30. 3.3 Projectile Motion Example #1:
Suppose the ball from the Figure in the previous slide is projected from a height of 25.0 m above the ground and is thrown with an initial horizontal velocity of 8.25 m/s.
A.) How long is the ball in flight before striking the ground?
B.) How far from the building does the ball strike the ground?

31. 3.3 Projectile Motion
A projectile launched in an arbitrary direction may have initial velocity components in both the horizontal and vertical directions, but its acceleration is still downward.
© 2010 Pearson Education, Inc.

32. 3.3 Projectile Motion
Look at arbitrary angle with golfer in previous slide.
The initial velocity is resolved into its components:
Vxo =
Vyo =
Vx =
Vy =

33. Question 3.5 Dropping a Package
a) quickly lag behind the plane while falling
b) remain vertically under the plane while falling
c) move ahead of the plane while falling
d) not fall at all
You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:
Answer: b

34. Question 3.6a Dropping the Ball I
a) the “dropped” ball
b) the “fired” ball
c) they both hit at the same time
d) it depends on how hard the ball was fired
e) it depends on the initial height
From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first?
Answer: c

35. Question 3.6c Dropping the Ball III
a) just after it is launched
b) at the highest point in its flight
c) just before it hits the ground
d) halfway between the ground and the highest point
e) speed is always constant
A projectile is launched from the ground at an angle of 30°. At what point in its trajectory does this projectile have the least speed?
Answer: b

36. 3.3 Projectile Motion Example:
Suppose a golf ball is hit off the tee with an initial velocity of 30.0 m/s at an angle of 35 degrees to the horizontal (like Figure from 2 slides ago)
A.) What is the maximum height reached by the ball?
B.) What is its range?

37. 3.3 Projectile Motion
Consider two balls, both thrown with the same initial speed vo, but one at an angle of 45 degrees above the horizontal and one below. Determine whether, upon reaching the ground,
A.) The ball projected upward will have the greater speed.
B.) The ball projected downward will have the greater speed.
C.) Both balls will have the same speed.
© 2010 Pearson Education, Inc.

38. 3.3 Projectile Motion
The range of a projectile is maximum (if there is no air resistance) for a launch angle of 45°.
© 2010 Pearson Education, Inc.

39. 3.3 Projectile Motion
With air resistance, the range is shortened, and the maximum range occurs at an angle less than 45°. [Reduced speed = reduced range]
© 2010 Pearson Education, Inc.

40. 3.3 Projectile Motion
A hockey player hits a “slap shot” in practice (with no goalie present) when he is 15.0 m directly in front of the net. The net is 1.20 m high, and the puck is initially hit an angle of 5.00 degree above the ice with a speed of 35.0 m/s.
A.) Determine whether the puck makes it into the net.

41. 3.4 Relative Velocity
Velocity is not an absolute quantity. Rather it is dependent upon the observer.
Its description is relative to the observer’s state of motion.
Motion of objects are described as being relative to Earth. Which we call a…)
Measurements are usually made with respect to some reference.

42. 3.4 Relative Velocity
Velocity may be measured in any inertial reference frame. At top, the velocities are measured relative to the ground; at bottom they are measured relative to the white car.
© 2010 Pearson Education, Inc.

43. 3.4 Relative Velocity
In two dimensions, the components of the velocity, and therefore the angle it makes with a coordinate axis, will change depending on the point of view.
© 2010 Pearson Education, Inc.

44. Review of Chapter 3
Two-dimensional motion is analyzed by considering each component separately. Time is the common factor.
© 2010 Pearson Education, Inc.

45. Review of Chapter 3 Vector components:
In projectile motion, the horizontal and vertical motions are determined separately.
© 2010 Pearson Education, Inc.

46. Review of Chapter 3 Range is the maximum horizontal distance traveled.
Relative velocity is expressed relative to a particular reference frame.
© 2010 Pearson Education, Inc.