So far, we have developed several tools to help us explore motion. First, there are a few basic mathematics relations. Other than the rules of algebra, we have developed tools for calculating with vectors, we have used some formulas for calculating areas, and we have explored the relationships among graphs used for studying motion.
VECTORS
For vectors, there are three ways of representing a vector:
1. graphically, as an arrow of a certain length in a certain direction.
2. polar form, as a magnitude or length and an angle
3. component form, listing the components in the directions of axes like the coordinates of a point.
Components of a vector
V in polar form (V,

):
V x
= V cos(

)
V y
= V sin(

) giving
V
= [V x
, V y
] = V x x
+ V y y
Putting the components back together:
Pythagorean theorem: a 2 + b 2 = c 2
Angle:

= tan -1 V y
/V x
Adding vectors:
1. Graphically: Draw the vectors to be added head to tail, maintaining their lengths and angles. The vector from the tail of the first vector to the head of the last is the sum of the vectors. To subtract a vector, reverse its direction and add it.
2. By Component: Resolve the vector into components as above. To add vectors, add their x components to get the x component of the sum and add the y components to get the y component.
One odd property of vectors is that they are not tied to the origin. The vector has a definite length and angle (direction), but not always a definite position.
Areas :
We used area calculations to work with velocity graphs. The basic formulae which will arise in calculating areas are: rectangle: w x h triangle: ½ b x h circle:
 r 2

Graphs:
For this class, we will restrict the acceleration graphs to horizontal lines (uniform acceleration). Since the acceleration is the slope of the velocity graph, this gives an upward slope for positive acceleration and downward slope for negative acceleration.
Velocity :
+ increasing + decreasing -- increasing -- decreasing
Position:
increasing + slope decreasing + slope increasing - slope decreasing - slope
Positive velocity = upward parabola Negative velocity = downward
Increasing speed = shallow to steep slope on parabola
Decreasing speed = steep to shallow parabola
----------------- area -------------------> acceleration velocity position
<------------------ slope ------------------
Note that, to get the velocity graph from the acceleration graph, you must know the initial velocity; and, to get the position graph from the velocity graph, you must know the initial position. Don’t confuse slope with coordinate position.
Then we actually developed some physics tools. We have been working out a model of motion. The basic equations in the model can be used for motion in any direction.
These are

MOTION (in any direction)
Constant velocity: a = 0 d = v t
Constant acceleration: v = v o
+ a t
V avg
= Δd/Δt = ½ (V o
+ V f
) d = d
0
+ v o
t + 1/2 a t v f
2 = v o
2 + 2 a
 d
2
These may be applied in the y direction, x direction, down a ramp, or any other linear direction for which you have the acceleration, displacement or velocity components in the direction of motion.
By selecting those for which you have data and combining them, you can solve motion problems in single or multiple directions. This is a bit like a jigsaw puzzle and may require some trial and error until you have more experience. The set of equations here will solve the problem for you.
Splitting problems:
Often problems must be split in order to solve them. Situations requiring a problem to be split are:
1. When the acceleration is different for different parts of the problem.
2. When there are more than one dimension to the problem.
3. When conditions change, such as shape of track, collision, and so on. These are really a subset of condition 1 above, since they involve a change in acceleration.
Implied known quantities: g = 9.80 m/s 2 at rest, from rest, ... V i
= 0 comes to a stop ... V f
= 0
Trajectory over flat terrain:
V yf
= -V yi
V y
= 0 at top t up = t down max horizontal range for angle of 45°

A general problem solving process:
1. Draw a diagram or sketch and label it with the values given for the variables.
2. Identify any implied values, like g, v=0, etc. and place them on the sketch.
3. Define a variable for the answer sought and place it on the sketch.
4. See if the problem needs to be split and do so, if needed. Look for a change in acceleration, 2 dimensions, .....
5. Check the set of equations to see which may be useful in connecting the given quantities to the answer sought.
6. Solve the equations.
7. Evaluate the answer to see if it is reasonable.
Ways to study this material.
Become very familiar with the information on these pages. You should strive to be able to write down and understand these relations at any time without referring to a text or notes.
Play with combining the relations enough that you can combine them easily to eliminate or solve for variables.
When deciding which equations are most likely helpful, look at what known values you have and what you want to calculate. Find as small a set of equations as possible involving the known values and the quantity to be calculated. Then take some time and paper (or board space) and play with them to create a solution
to the problem.
It’s not a matter of remembering how to solve a problem, you use the basic tools to create a solution, whether or not you have seen such a solution before.