**Where are we going with this?**The information on this page relates to the skills needed to investigate and evaluate the graphical and mathematical relationship (using either manual graphing or computers) of one-dimensional kinematic parameters (distance, displacement, speed, velocity, acceleration) with respect to an object's position, direction of motion, and time.

**Overviewing Vectors**

*(Oh brother! What now?)*

**vector**

*is a concept connected to quantities that specifies a magnitude and a direction.*Answering many physics questions about the world around us best done by using vectors.

**Understanding by example…**

The box moved 20 m to the left.

*∆x*) by a magnitude of 20 m and in the direction of left.

**Working in 2D motion, the concept of vectors was assumed when positive and negative values were assigned to thing happening in opposite directions.**When motion was to the right and acceleration was to the left, their magnitudes were given different signs. Likewise for up and down.

*EXAMPLE 1*

*Working with Directions on the SAME line.*

*A ball with an initial velocity of 9 m/s is thrown directly upward on a planet where acceleration due to gravity is 3 m/s*

^{2}. How long will it rise before stopping?Find tWherevi = 9 m/sa = 3 m/s^{2}vf = 0 m/sand wherevf = vi + at

Find tWherevi = 9 m/s upa = 3 m/s^{2 }down

vf = 0 m/s up

and wherevf = vi + at

*It's a nit that we usually do mentally, but…*3rd round…

Find tWhere up is positive

vi = 9 m/s upa = 3 m/s^{2 }down = -3 m/s^{2 }up

vf = 0 m/s up

and wherevf = vi + at

vf = vi + at0 = 9 m/s + (-3 m/s^{2})(t)-9 m/s = (-3 m/s^{2})(t)3 s = t

**Moving on into 2 dimensions**, we'll stick with displacement as the quantity…

*EXAMPLE 2*

*Pizza Delivery Route as Vectors*

*point A*is some dude or dudette who wants to get to

*point B*. To do so, travel must result in TWO displacements (

*ish*… Just hang in here as we develop the concept.)

1 block right…2 blocks up…1 block right…1 block up…1 block right.

*∆P*

*∆P = ∆x + ∆y*(ish, not exactly, but we're getting there…

*∆x*is how much did it move on the

*x*axis.

*∆y*is how much did it move on the

*y*axis.

2 blocks up1 block up1 block right1 block right1 block right

So, up isyand right isx…

∆x= 1 block + 1 block + 1 block∆y= 2 blocks + 1 block

∆P= 3 blocks up + 3 blocks rightor∆P= 3 blocks(y) + 3 blocks(x)

*What just happened?*

**We added vectors.**

*x*vectors and added them, then we took all of the

*y*vectors and added them. Bam!

*Okay, so… what if we wanted to get a little more formal?*

Component vector:Any individual vector having magnitude and direction within a given frame of reference.Component vectors can be combined into aresulting vector(or theresultant vector).

What if you start with the resultant? Yeah, that can happen!

A resultant vector can be broken down into its component vectors.

**Now, some caveats… common sense that needs to be outed…**1.Within a frame of reference(for simplicity, let's say thex-y axisframe of reference),component vectors have magnitudes ONLY on one axis.2. The frame of reference will assign positive and negative to some direction on each axis; usually, up on the y axis is positive and right on the x axis is positive.

fig 3 |

3. The resultant vector can be described as the sum of thexcomponents and the sum of theycomponents.

4. However, it can also be described by its own (resultant) magnitude and an angle with respect to one of the axes (frequently, with respect to thexaxis).

Returning to the pizza deliver (fig 3) robot, the

blue line represents the resultant vector.While we can say that the total displacement is…

3 blocks right and 3 blocks up

we could also go with the x-y axis thing and say…

∆x = 3; ∆y = 3

Woh… that got mathy fast!or we could say:

The resultant vector is some magnitude at some angle to the x axis.

Using the Pythagorean theorem, the blue line has a length that is the square root of 18. Geometry tells us that the angle of a triangle with two equal sides is 45°.

So…the resulting vector is

√18 = blocks long in a direction that is 45° above the x axis.

Enough, already… really! What happened to the pizza?

5.

When "doing the math" you can't combine vectors of different quantities. For instance, you can't combine velocity vectors with acceleration vectors directly. They operate on an object differently.

6. Many different paths to deliver the pizza can end up at the same place.The robot could go 10 to the right, 10 up, then 7 left and 7 down. The sum of these component vectors would yield the same displacement:

∆x = 3; ∆y = 3

*EXAMPLE 3*

*The 1100 mile per hour pitch.*

**From the frame of reference where the sun is not moving, then the velocity of the baseball (if thrown with the rotation of the earth is the sum of the pitch plus the velocity of a point on the surface of the earth.**

**Each of those component velocities**

*could*be added together to find the resultant velocity of the baseball.

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