1. Kinematics in Two or Three Dimensions; Vectors
Chapter 3
Kinematics in Two or Three Dimensions; Vectors
Chapter Opener. Caption: This snowboarder flying through the air shows an example of motion in two dimensions. In the absence of air resistance, the path would be a perfect parabola. The gold arrow represents the downward acceleration of gravity, g. Galileo analyzed the motion of objects in 2 dimensions under the action of gravity near the Earth’s surface (now called “projectile motion”) into its horizontal and vertical components. We will discuss how to manipulate vectors and how to add them. Besides analyzing projectile motion, we will also see how to work with relative velocity.

2. 3-1 Vectors and Scalars A vector has magnitude as well as direction.
Some vector quantities: displacement, velocity, force, momentum
A scalar has only a magnitude.
Some scalar quantities: mass, time, temperature
Figure 3-1. Caption: Car traveling on a road, slowing down to round the curve. The green arrows represent the velocity vector at each position.

3. 3-2 Addition of Vectors—Graphical Methods
For vectors in one dimension, simple addition and subtraction are all that is needed.
You do need to be careful about the signs, as the figure indicates.
Figure 3-2. Caption: Combining vectors in one dimension.

4. 3-2 Addition of Vectors—Graphical Methods
If the motion is in two dimensions, the situation is somewhat more complicated.
Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.
Figure 3-3. Caption: A person walks 10.0 km east and then 5.0 km north. These two displacements are represented by the vectors D1 and D2, which are shown as arrows. The resultant displacement vector, DR, which is the vector sum of D1 and D2, is also shown. Measurement on the graph with ruler and protractor shows that DR has a magnitude of 11.2 km and points at an angle θ = 27° north of east.

5. 3-2 Addition of Vectors—Graphical Methods
Adding the vectors in the opposite order gives the same result:
Figure 3-4. Caption: If the vectors are added in reverse order, the resultant is the same. (Compare to Fig. 3–3.)

6. 3-2 Addition of Vectors—Graphical Methods
Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.
Figure 3-5. Caption: The resultant of three vectors: vR = v1 + v2 + v3.

7. 3-2 Addition of Vectors—Graphical Methods
The parallelogram method may also be used; here again the vectors must be tail-to-tip.
Figure 3-6. Caption: Vector addition by two different methods, (a) and (b). Part (c) is incorrect.

8. 3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar
In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.
Then we add the negative vector.
Figure 3-7. Caption: The negative of a vector is a vector having the same length but opposite direction.
Figure 3-8. Caption: Subtracting two vectors: v2 – v1.

9. 3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar
A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.
Figure 3-9. Caption: Multiplying a vector v by a scalar c gives a vector whose magnitude is c times greater and in the same direction as v (or opposite direction if c is negative).

10. 3-4 Adding Vectors by Components
Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.
Figure Caption: Resolving a vector V into its components along an arbitrarily chosen set of x and y axes. The components, once found, themselves represent the vector. That is, the components contain as much information as the vector itself.

11. 3-4 Adding Vectors by Components
Remember:
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If the components are perpendicular, they can be found using trigonometric functions.
Figure Caption: Finding the components of a vector using trigonometric functions.

12. 3-4 Adding Vectors by Components
The components are effectively one-dimensional, so they can be added arithmetically.
Figure Caption: The components of v = v1 + v2 are
vx = v1x + v2x
vy = v1y + v2y

13. 3-4 Adding Vectors by Components
Draw a diagram; add the vectors graphically.
Choose x and y axes.
Resolve each vector into x and y components.
Calculate each component using sines and cosines.
Add the components in each direction.
To find the length and direction of the vector, use:
and .

14. 3-4 Adding Vectors by Components
Example 3-2: Mail carrier’s displacement.
A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?
Figure Caption: Example 3–2. (a) The two displacement vectors, D1 and D2. (b) D2 is resolved into its components. (c) D1 and D2 are added graphically to obtain the resultant D. The component method of adding the vectors is explained in the Example.
Answer: The x and y components of her displacement are km and km.
The magnitude of her displacement is 30.0 km, at an angle of 38.5° below the x axis.

15. 3-4 Adding Vectors by Components
Example 3-3: Three short trips.
An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement?
Figure 3-14.
Answer: The x and y components of the displacement are 600 km and -750 km.
Alternatively, the magnitude of the displacement is 960 km, at an angle of 51° below the x axis.

16. 3-5 Unit Vectors Unit vectors have magnitude 1.
Using unit vectors, any vector can be written in terms of its components:
Figure Caption: Unit vectors i, j, and k along the x, y, and z axes.

17. 3-6 Vector Kinematics
In two or three dimensions, the displacement is a vector:
Figure Caption: Path of a particle in the xy plane. At time t1 the particle is at point P1 given by the position vector r1; at t2 the particle is at point P2 given by the position vector r2. The displacement vector for the time interval t2 – t1 is Δr = r2 – r1.

18. 3-6 Vector Kinematics
As Δt and Δr become smaller and smaller, the average velocity approaches the instantaneous velocity.
Figure Caption: (a) As we take Δt and Δr smaller and smaller [compare to Fig. 3–16] we see that the direction of Δr and of the instantaneous velocity (Δr/ Δt, where Δt →0) is (b) tangent to the curve at P1 .

19. 3-6 Vector Kinematics
The instantaneous acceleration is in the direction of Δ = 2 – 1, and is given by:
Figure Caption: (a) Velocity vectors v1 and v2 at instants t1 and t2 for a particle at points P1 and P2, as in Fig. 3–16. (b) The direction of the average acceleration is in the direction of Δv = v2 – v1.

20. 3-6 Vector Kinematics
Using unit vectors,

21. 3-6 Vector Kinematics
Generalizing the one-dimensional equations for constant acceleration: