Unit 2: Vectors and 2 DimensionsName ______

Vectors:

What is 2 + 2? _____ Is it always?

Example: A student walks 2 miles north, then walks 2 miles east.

  1. What distance has he walked?
  1. What is his displacement? (Displacement is position relative to his starting point.)

For this example, you were asked to find distance, which is a scalar, and displacement, which is a vector.

Scalars are quantities which have only ______(size). They are normal numbers, and you can just add them.

Vectors are quantities that have both ______and ______. You can’t simply add them, but must take the directions into account.

Representing Vectors:

Vectors are represented as ______. The ______of the vector is represents its magnitude.

Practice:

  1. In the space below, from a common origin, draw the following vectors to a scale of 5 meters = 1 cm.
  2. 20 meters, west.
  3. 40 meters, north.
  4. 25 meters, 400 north of west.

Adding Vectors:

  1. Sketch the situation.
  2. Move the vectors “tip to tail”.
  3. Draw in the resultant from the origin to the tip of the last vector.
  4. Find the magnitude and direction of the resultant by:
  5. Scale model
  6. Using components (More on this later)
  7. Law of Cosines  c2 = a2 + b2- 2ab∙cos(c)

Example: Two forces with magnitudes of 20 Newtons and 30 Newtons (newtons is the unit for force) act on an object. Find the resultant force for each situation.

  1. The two forces are both pointing East.
  1. 20 Newtons east and 30 Newtons west.
  1. 20 Newtons east and 30 Newtons north.
  1. 20 Newtons east and 30 Newtons, 400north of east…

Finding Components of Vectors:

Components = ______. The components of a vector must add to form the vector.

Example: For the vector below, A and B are components. So are C and D, and E and F.

How many sets of components are there for any vector?

Horizontal (x) and vertical (y) components:

Two components of a vector will be useful, the x and y components.

To find the components, notice that Ax is the adjacent side of the triangle, and A the hypotenuse. We can use cosine to relate them:

Cos(q) = AxWhich can be rewritten as Ax = Acos()

A

Similarly, we can find Ay by using sine. Ay = Asin()

Example: For the problems below, find the horizontal and vertical components:

Adding Vectors Using Components:

  1. Find the x and y components of all vectors.
  2. Find the components of the resultant by adding the individual components.

(Rx = Ax + Bx + … and Ry = Ay + By + … )

  1. Construct the vector R using Rx and Ry

Example:

Find the resultant for 20 Newtons east and 30 Newtons, 400north of east…

2 Dimensional Motion

Split into 2 parts, x and y!

Example:

A boat travels at a speed of 8 m/s, east across a river flowing at 6 m/s, south.

Projectile Motion:

Definitions:

A projectile is an object that is only under the influence of ______(once it is fired).

The trajectory is the ______the projectile takes.

Example: If a bullet is fired from a gun, and a bullet is dropped from the same height at the same time, which will hit the ground first?

The x- and y- motions are ______of one another.

For all cases, ay = ______, ax = ______

Case 1: Projectile fired horizontally: (V0y = 0)

Example: A ball is thrown horizontally with a speed of 12 m/s off a 20 meter tall building…

Case 2: Projectile fired at an angle:

Example: A soccer ball is kicked with a speed of 20 m/s at an angle of 150 above horizontal.

Split v0 into v0x and v0y:

Facts you should know…

  1. Acceleration is ______at all points along trajectory.
  1. Velocity is ______.
  1. Time depends on the ______motion. Time is ______for x and y.
  1. The farthest possible distance is for a projectile launched at ______.

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