A frame of reference is an object or set of objects relative to which motion is measured.

It is the object that you choose to consider as unmoving. In most cases we choose the earth as our frame of reference, even though it is most certainly in motion.

How fast is the Earth moving? It depends on what you choose as your frame of reference.

A vector is a measurement that has both an amount and a direction associated with it.

Velocity is a vector quantity. Acceleration and force are also vectors.

When adding vectors you must consider the direction of each.

The net velocity is the total of two or more combined velocities.

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Examples: 20 m/s + 30 m/s = 50 m/s but 20 m/s + 30 m/s = 10 m/s

Here is how to solve a vector problem:

First, find the net velocity, then use that in your velocity equation.

Example:

A person runs up an escalator at 5 m/s, as the escalator moves down at 3 m/s, how long will it take the person to reach the top of the escalator if it is 18 meters long?

Find the net velocity: 5 m/s + ¯ 3 m/s = 2 m/s

Solve for time using net velocity: 18m ¸ 2 m/s = 9 seconds

Remember, if both objects are traveling together in the same frame of reference, then the motion of the frame of reference can be ignored.

Example:

Person “A” runs up an escalator at 5 m/s, as the escalator moves down at 3 m/s.

Person “B” is 6 meters ahead of the first person, and walking up the escalator at

1 m/s. How long until they collide?

Find the net velocity: 5 m/s – 1 m/s = 4 m/s (ignore the motion of the escalator)

Solve for time using net velocity: 6 m ¸ 4 m/s = 1.5 seconds

In general we can use the following guidelines for vector problems:

If two objects are moving independently of each other (ex: two cars)

If one object is traveling across a moving surface (ex: person on escalator) or through a moving substance (ex: plane through the air, boat through water)

a) motion of both are in same direction ® ® add (ex: boat is moving with the current)

b) motion is in opposite directions ® ¬ subtract (ex: person walks up the down escalator)

The rules above do not apply when the objects are moving at angles to one another

The graphical method of vector addition

Remember, a vector is a quantity that has both an amount and a direction.

For the purposes of this class we will always use the right side of the paper as 0^o, the top as 90^o, the left as 180^o, and the bottom as 270^o

We will use arrows to represent vectors.

The direction of the vector is the direction the arrow points and can be found using a protractor.

The amount of the vector is represented by its length, which we will always measure in centimeters.

Examples: 1 cm = 10 m/s

(you decide what 1 cm will equal)

8 m/s at 0^o 18 m/s at 90^o 39 m/s at 40^o 23 m/s at 290^o

When we add two or more vectors, the outcome is called the resultant vector.

One way to find the resultant vector is to make a small “x” at your starting point, and then draw an arrow representing the first vector. Remember that the length of this arrow represents its amount, and its direction is the direction the arrow points.

Draw the second vector arrow starting from the tip of the first arrow.

If there is a third vector, start it at the tip of the second vector.

To find the resultant vector (net velocity), draw an arrow starting at the same point as the first arrow and ending at the tip of the last vector arrow.

The length of this arrow is the magnitude of the vector (not the distance traveled!).

The direction of the vector is determined with a protractor.

Examples: a bird flies North at 15 m/s as the wind blows it east at 28 m/s . What is the bird’s net velocity?

You want your boat to go due North (90^o). Its throttle is set at 34 m/s. If the water is moving 31 m/s in a direction of 0^o (due East) , what direction should you aim for? How fast will its northward progress be?

Hint make a resultant vector line that points north and see where the boat’s vector crosses the line of the resultant vector

Boat must head at 154^o and will move

Northward at 15 m/s

It is often useful to consider only the “x” component or “y” component of a vector. This allows us to separate vertical from horizontal, or north/south from east/west.

Example: using a scale of 1 cm = 10 km/hr the arrow below represents a velocity vector

With a magnitude of 38 km/hr and a direction of 300^o

We can resolve this vector into its x and y components by measuring the amount of displacement that occurred along the x axis and the y axis.

x component = 19 km/hr

y component = 33 km/hr

Another way to find the value of the x and y components of a vector is to use the

sine and cosine buttons on your calculator. (sin and cos)

To find the y component, enter the angle (in this case 300^o), and then hit the sine button. Multiply this number by the magnitude of the vector (in this case 38).

Sine (300) ´ 38 km/hr = -32.9 km/hr the negative value indicates that the y component

is pointing in the negative y direction (down the page)

To find the x component, enter the angle (in this case 300^o), and then hit the cosine button.

Multiply this number by the magnitude of the vector (in this case 38).

Cosine (300) ´ 38 km/hr = 19 km/hr The positive value indicates that the x component is

Pointing in the positive x direction (right)