Kinematics & Dynamics in 2 & 3 Dimensions; Vectors First, a review of some Math Topics in Ch. 1. Then, some Physics Topics in Ch. 4!

Vectors: Some Topics in Ch. 1, Section 7 General Discussion. Vector  A quantity with magnitude & direction. Scalar  A quantity with magnitude only. Here, we’ll mainly deal with Displacement & Velocity. But, our discussion is valid for any vector! The Ch. 1 vector review has a lot of math! It requires a detailed knowledge of trigonometry! Problem Solving A diagram or sketch is helpful & vital! I don’t see how it is possible to solve a vector problem without a diagram!

Coordinate Systems Rectangular (Cartesian) Coordinates “Standard” coordinate axes. A point in the plane is (x,y) If its convenient, we could reverse + & - -,++,+ -, -+, - A “Standard Set” of xy Coordinate Axes

Vector & Scalar Quantities Vector  Quantity with magnitude & direction. Scalar  Quantity with magnitude only. Equality of Two Vectors Consider 2 vectors, A & B A = B means A & B have the same magnitude & direction.

Vector Addition, Graphical Method Addition of Scalars: We use “Normal” arithmetic! Addition of Vectors: Not so simple! Vectors in the same direction: –We can also use simple arithmetic Example 1: Suppose we travel 8 km East on day 1 & 6 km East on day 2. Displacement = 8 km + 6 km = 14 km East Example 2: Suppose we travel 8 km East on day 1 & 6 km West on day 2. Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement

Adding Vectors in the Same Direction:

Graphical Method of Vector Addition For 2 vectors NOT along the gsame line, adding is more complicated: Example: D 1 = 10 km East D 2 = 5 km North. What is the resultant (final) displacement? 2 Methods of Vector Addition: –Graphical (2 methods of this also!) –Analytical (TRIGONOMETRY)

Graphical Method of Adding Vectors “Recipe” Draw the first vector. Draw the second vector starting at the tip of the first vector Continue to draw vectors “tip-to-tail” The sum is drawn from the tail of the first vector to the tip of the last vector Example:

Example: 2 vectors NOT along the same line. Figure! D 1 = 10 km E, D 2 = 5 km N. Resultant = D R = D 1 + D 2 = ? In this special case ONLY, D 1 is perpendicular to D 2. So, we can use the Pythagorean Theorem. D R = 11.2 km Note! D R < D 1 + D 2 (scalar addition )

D 1 = 10 km E, D 2 = 5 km N. Resultant = D R = D 1 + D 2 = ? The Graphical Method of Addition Plot the vectors to scale, as in the figure. Then measure D R & θ. Results in D R = 11.2 km, θ = 27º N of E D R = 11.2 km Note! D R < D 1 + D 2

This example illustrates general rules of graphical addition, which is also called the “Tail to Tip” Method. Consider R = A + B (See figure!). Graphical Addition Recipe 1. Draw A & B to scale. 2. Place the tail of B at the tip of A 3. Draw an arrow from the tail of A to the tip of B 4. This arrow is the Resultant R Measure its length & the angle with the x-axis.

Order Isn’t Important! Adding vectors in the opposite order gives the same result: In the example in the figure, D R = D 1 + D 2 = D 2 + D 1

Graphical Method of Vector Addition Adding (3 or more) vectors: V = V 1 + V 2 + V 3 Even if the vectors are not at right angles, they can be added graphically using the tail-to-tip method.

A 2nd Graphical Method of Adding Vectors (equivalent to the tail-to-tip method, of course!) V = V 1 + V 2 1. Draw V 1 & V 2 to scale from a common origin. 2. Construct a parallelogram using V 1 & V 2 as 2 of the 4 sides. 3. Resultant V = Diagonal of the Parallelogram from a Common Origin (measure length & the angle it makes with the x axis) See Figure Next Page! Parallelogram Method

Mathematically, we can move vectors around (preserving their magnitudes & directions) A common error! Parallelogram Method

Subtraction of Vectors First, Define The Negative of a Vector: - V  vector with the same magnitude (size) as V but with opposite direction. Math: V + (- V)  0 Then add the negative vector. For 2 vectors, V 1 & V 2 :

Subtracting Vectors To subtract one vector from another, add the first vector to the negative of the 2 nd vector, as in the figure below:

Multiplication by a Scalar A vector V can be multiplied by a scalar c V' = cV V'  vector with magnitude cV & same direction as V. If c is negative, the resultant is in the opposite direction.

Example Consider a 2 part car trip: Displacement A = 20 km due North. Displacement B = 35 km 60º West of North. Find (graphically) resultant displacement vector R (magnitude & direction). R = A + B. See figure below. Use ruler & protractor to find the length of R & the angle β. Answers: Length = 48.2 km β = 38.9º