1. Section 2.7 Free Fall
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2. Free Fall
If an object moves under the influence of gravity only, and no other forces, we call the resulting motion free fall.
Any two objects in free fall, regardless of their mass, have the same acceleration.
On the earth, air resistance is a factor. For now we will restrict our attention to situations in which air resistance can be ignored.
Apollo 15 lunar astronaut David Scott performed a classic experiment on the moon, simultaneously dropping a hammer and a feather from the same height. Both hit the ground at the exact same time—something that would not happen in the atmosphere of the earth!
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3. Free Fall
The figure shows the motion diagram for an object that was released from rest and falls freely. The diagram and the graph would be the same for all falling objects.
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4. Free Fall
g, by definition, is always positive. There will never be a problem that uses a negative value for g.
Even though a falling object speeds up, it has negative acceleration (–g).
Because free fall is motion with constant acceleration, we can use the kinematic equations for constant acceleration with ay = –g.
g is not called “gravity.” g is the free-fall acceleration.
g = 9.80 m/s2 only on Earth’s surface. Other planets have different values of g, and g is different at different heights.
We will sometimes compute acceleration in units of g.
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5. QuickCheck 2.26
A ball is tossed straight up in the air. At its very highest point, the ball’s instantaneous acceleration ay is
Positive.
Negative.
Zero.
Answer: B
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6. QuickCheck 2.26
A ball is tossed straight up in the air. At its very highest point, the ball’s instantaneous acceleration ay is
Positive.
Negative.
Zero.
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7. Chapter 3 Vectors and Motion in Two Dimensions
Chapter Goal: To learn more about vectors and to use vectors as a tool to analyze motion in two dimensions.
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8. Chapter 3 Preview Looking Ahead
Text: p. 64
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9. Section 3.1 Using Vectors
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10. Using Vectors
A vector is a quantity with both a size (magnitude) and a direction.
Figure 3.1 shows how to represent a particle’s velocity as a vector .
The particle’s speed at this point is 5 m/s and it is moving in the direction indicated by the arrow.
The magnitude of a vector, a scalar quantity, cannot be a negative number
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11. Using Vectors
The displacement vector is a straight-line connection from the initial position to the final position, regardless of the actual path.
Two vectors are equal if they have the same magnitude and direction. This is regardless of the individual starting points of the vectors.
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12. Vector Addition
is the net displacement because it describes the net result of the hiker’s having first displacement , then displacement .
The net displacement is an initial displacement plus a second displacement :
The sum of the two vectors is called the resultant vector. Vector addition is commutative:
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13. Vector Addition
The figure shows the tip-to-tail rule of vector addition and the parallelogram rule of vector addition – you can use either of these methods to add vectors – they give the same answer!
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14. QuickCheck 3.1 Given vectors and , what is ? Answer: A
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15. QuickCheck 3.1 Given vectors and , what is ? A.
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16. Multiplication by a Scalar
Multiplying a vector by a positive scalar gives another vector of different magnitude but pointing in the same direction (provided c > 0).
If we multiply a vector by zero the product is a vector having zero length - the zero vector!
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17. Multiplication by a Scalar
A vector cannot have a negative magnitude.
If we multiply a vector by a negative number we reverse its direction.
Multiplying a vector by –1 reverses its direction without changing its length (magnitude).
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18. QuickCheck 3.2 Which of the listed vectors is ? Answer: C
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19. QuickCheck 3.2 Which of the listed vectors is ? C.
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20. Vector Subtraction
Text: p. 67
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21. QuickCheck 3.3 Given vectors and , what is ? Answer: D
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22. QuickCheck 3.3 Given vectors and , what is ? D.
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23. QuickCheck 3.4 Which of the vectors listed is 2  ? Answer: A
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24. QuickCheck 3.4 Which of the vectors listed is 2  ? A.
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25. Section 3.2 Using Vectors on Motion Diagrams
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26. Using Vectors on Motion Diagrams
In two dimensions, an object’s displacement is a vector:
The (average) velocity vector is simply the displacement vector multiplied by the scalar 1/Δt – this is the same as the instantaneous velocity vector if the motion is uniform.
Generally, this is approximately the instantaneous velocity vector as Δt becomes very small.
Consequently the velocity vector points in the direction of the displacement.
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27. Example 3.1 Finding the velocity of an airplane
A small plane is 100 km due east of Denver. After 1 hour of flying at a constant speed in a constant direction, it is 200 km due north of Denver. What is the plane’s velocity?
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28. Example 3.1 Finding the velocity of an airplane
A small plane is 100 km due east of Denver. After 1 hour of flying at a constant speed in a constant direction, it is 200 km due north of Denver. What is the plane’s velocity? prepare The initial and final positions of the plane are shown in FIGURE 3.8; the displacement is the vector that points from the initial to the final position.
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29. Example 3.1 Finding the velocity of an airplane (cont.)
solve The length of the displacement vector is the hypotenuse of a right triangle: The direction of the displacement vector is described by the angle  in Figure 3.8. From trigonometry, this angle is
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30. Example 3.1 Finding the velocity of an airplane (cont.)
Thus the plane’s displacement vector is Because the plane undergoes this displacement during 1 hour, its velocity is assess The plane’s speed is the magnitude of the velocity, v = 224 km/h. This is approximately 140 mph, which is a reasonable speed for a small plane.
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31. Acceleration Vectors
The vector definition of (average) acceleration is a straightforward extension of the one-dimensional version:
There is an acceleration whenever there is a change in velocity. Velocity can change in either or both of two possible ways:
The magnitude can change, indicating a change in speed.
The direction of motion can change.
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