Vectors & Scalars Physics 11

Vectors & Scalars A vector has magnitude as well as direction. Examples: displacement, velocity, acceleration, force, momentum A scalar has only magnitude Examples: time, mass, temperature

Vector Addition – One Dimension A person walks 8 km East and then 6 km East. Displacement = 14 km East A person walks 8 km East and then 6 km West. Displacement = 2 km

Vector Addition Example 1: A person walks 10 km East and 5.0 km North Order doesn’t matter

Graphical Method of Vector Addition Tail to Tip Method

Graphical Method of Vector Addition “Head-to-Tail” Method

Graphical Method of Vector Addition Parallelogram Method Helpful hints about parallelograms: All four angles add to equal 360 o Opposite angles are equal

Properties of Parallel Lines

Subtraction of Vectors Negative of vector has same magnitude but points in the opposite direction. For subtraction, we add the negative vector.

Multiplication by a Scalar A vector V can be multiplied by a scalar c The result is a vector cV that has the same direction but a magnitude cV If c is negative, the resultant vector points in the opposite direction.

Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components (i.e. V x & V y ). Components are chosen so that they are perpendicular to each other.

Trigonometry Review Opposite Adjacent Hypotenuse Pythagorean Theorem: (Hypotenuse) 2 = (Opposite) 2 + (Adjacent) 2

Adding Vectors by Components If the components are perpendicular, they can be found using trigonometric functions.

Adding Vectors by Components The components are effectively one-dimensional, so they can be added arithmetically:

Signs of Components

Adding Vectors by Components Adding vectors: 1. Draw a diagram; add the vectors graphically. 2. Choose x and y axes. 3. Resolve each vector into x and y components. 4. Calculate each component using sines and cosines. 5. Add the components in each direction. 6. To find the length and direction of the vector, use:

Relative Velocity Relative velocity considers how observations made in different reference frames are related to each other. Example: A person walks toward the front of a train at 5 km/h (V PT ). The train is moving 80 km/h with respect to the ground (V TG ). What is the person’s velocity with respect to the ground (V PG )?

Relative Velocity Boat is aimed upstream so that it will move directly across. Boat is aimed directly across, so it will land at a point downstream. Can expect similar problems with airplanes.

Practise Problems #1, page 70 #9, page 71 #12, page 71 #40, page 73 #41, page 74