1. Kinematics Vectors and Motion

2. Objectives
Describe how a vector is altered when multiplied by a scalar, and also what aspect is not altered.
Explain how the orientation of the acceleration vector affects velocity.
Draw the final velocity vector given the initial velocity and acceleration vectors
Draw the acceleration vector given the initial and final velocity vectors
Determine vector components
Resolve vector components
Add vectors that are NOT parallel or perpendicular, using components, to determine the magnitude and direction of the resultant

3. Multiplying a Vector by a Scalar
Multiplying a vector by a scalar results in a new vector
Magnitude of the new vector:
The product of the original vector’s magnitude and the scalars magnitude.
Direction of the new vector:
Direction is the same if the scalar is positive.
Direction is the opposite if the scalar is negative.
Examples:
Visual:
Equation:
s = v t Multiplying the velocity vector by the scalar time results in the displacement vector. A 2A
 2A

4. Curving Motion and the Instantaneous Velocity Vector
If an object follows a curved trajectory the velocity vector at any instant is always drawn tangent to trajectory.
Orientation of Acceleration Vector and Affect on Velocity
When the acceleration vector and velocity vector are parallel and point in the same direction, then the object’s speed increases.
When the acceleration vector and velocity vector are parallel and point in the opposite directions, then the object’s speed decreases.
When the acceleration vector and velocity vector are perpendicular, then the object’s direction changes.
What if they are at slight angles to each other?

5. Orientation of Acceleration Vector and Affect on Velocity
When the acceleration vector and velocity vector are parallel and point in the same direction, then the object’s speed increases.
When the acceleration vector and velocity vector are parallel and point in the opposite directions, then the object’s speed decreases.
When the acceleration vector and velocity vector are perpendicular, then the object’s direction changes.
What if they are at slight angles to each other?

6. Orientation of Acceleration Vector and Affect on Velocity
Acceleration vector is diagonal: Split it into components
The y-component is in the same direction as velocity, and it increase the object’s speed.
The x-component is perpendicular to velocity, and it will cause the object to turn to the right.
Speed is increasing and this object is turning to the right.
The y-component is opposite of velocity, and it decrease the object’s speed.
Speed is decreasing and this object is turning to the right. v a ay ax v a ax ay

7. Visual Relationship: Velocity and Acceleration Vectors
In the vector version of the velocity-time
v = v0 + a t
Multiplying the acceleration vector by the time scalar results in a change in velocity vector during time t .
a t = v
This means that all three quantities in the velocity-time equation are like vectors with like units.
When we solve this equation mathematically, we select only vectors that are parallel to each other in order to do ordinary addition, without involving geometry and trigonometry. When this is done we drop the vector notation and treat these quantities as scalars.
Can we add vectors if they are not parallel?
YES. But, this is usually only done conceptually, and pictorially, on the AP Exam. How is this done?

8. Visual Relationship: Velocity and Acceleration Vectors
Suppose an object has an initial velocity of 4 m/s in the +y-direction and it is acted upon by an acceleration of 3 m/s2 in the +x-direction. Sketch the addition of these vectors and the resultant final velocity vector.
Draw the initial velocity. This first vector arrow sets the scale, and the length of subsequent vector arrows should be proportional to this vector.
Add the change in velocity (v = a t ) vector to the initial velocity vector.
The final velocity is the resultant: v = v0 + a t
Why when this done, in the text and AP Exam, is the a t vector frequently just written as the acceleration, a , vector?
If time equals one second, then a t = a (1) = a
When these vectors are added conceptually a time interval of one second is assumed.
a t v0 v

9. Visual Relationship: Velocity and Acceleration Vectors
What if the initial and final velocity vectors are given and the problem is asking for the approximate direction of the acceleration vector.
If we examine the previous problem, shown below, we see that when the velocity vectors are oriented tail to tail, the change in velocity, v = a t , vector points from the tip of the initial velocity vector toward the tip of the final velocity vector.
Let’s take a look at how this might appear on an exam.
a t v0 v

10. Visual Relationship: Velocity and Acceleration Vectors
An object moves at constant speed along the path shown below from point A to point B. Identify the direction of the average acceleration.
Orient the velocity vectors tip to tail. Simply move the final velocity vector.
Then draw the change in velocity vector, v = a t , from the tip of the initial velocity vector to the tip of the final velocity vector.
This method returns the average acceleration as the object moves from A to B. Instantaneous acceleration is continually changing during this motion. (More on this later).
NOTE: The object has an initial y-velocity and must slow down in the y-direction (ay) until it has zero y-velocity. In the x- direction the object has zero initial x- velocity, which has to increase in the x- direction (+ax) to reach its new final value. The components of average acceleration must be –y (slowing) and +x (speeding up). Average acceleration points down and right. v v0 B
a t A v

11. Vector Components
Suppose you encounter a problem where an object is launched at an angle above the ground with a velocity vector of 50 m/s at 37o,
but the problem is only interested in how fast the object is moving horizontally or vertically.
The given vector can be converted in components.
Finding components is the opposite of vector addition. You are trying to find the lengths of the horizontal and vertical vectors that equal the original vector when added together. v 
v = vx + vy

12. Vector Components v vy  vx
Components of vectors are labeled with the same variable letter. Then add subscripts, x or y , corresponding to the axis they lie along.
The horizontal component (width of v )is adjacent the given angle 
Magnitude: vx = hyp cos  vx = v cos 
Direction: Depends on which way it points, which in this case is +x
For mathematical purposes it will be recorded as a scalar where the sign indicates direction (+ = right) and ( = left) and magnitude its value.
vx = v cos  vx = 50 cos 37o vx = +40 m/s
The vertical component (height of v ) is opposite the given angle 
vy = v sin  vy = 50 sin 37o vy = +30 m/s

13. Vector Components 217o vx 37o vy v
What about other quadrants?
In the diagram at the right
37o is a reference angle 37o below the x axis
217o does not require reference since it is measured from +x
Use either angle to solve components,
However, if reference angles are used, then + and  signs representing direction must be inserted manually after completing the calculation.
vx = 50 cos 37o = 40 =  40 m/s vy = 50 sin 37o = 30 =  30 m/s
If angles measured from the +x axis (default 0o ) are used, then the sign representing direction is automatically calculated.
vx = 50 cos 217o =  40 m/s vy = 50 sin 217o = 30 =  30 m/s

14. Resolving Components V = ? vy = 30  = ? vx = 40
In some problems the components of a parent vector are given and you must determine the magnitude and direction of the parent vector.
The components lie along different axes and are therefore always perpendicular to one another.
The parent vector extends from the tail of the first vector (x component) to the tip of the second vector (y component)
The three vectors form a right triangle.
To determine the parent vectors magnitude use Pythagorean theorem.
To determine direction use inverse tangent. Caution: This calculates a reference angle. Either state the complete reference or determine the angle measured from the +x axis to avoid stating a reference.

15. Adding Vectors that are Parallel and Perpendicular
See the Math Tools presentation for more detail on these two easier types of vector addition
For parallel vectors
Ensure they have the correct sign depending on the direction they point.
Use ordinary addition
Note: these vectors do not have to lie on the x-axis or y-axis. Any parallel vectors can be added using ordinary addition.
Example: A 30o vector can be easily added to a 210o vector. The 210o is 180o from 30o, and is therefore parallel to the 30o vector. Set the vector at 210o as negative, and then just add its magnitude to the magnitude of the 30o vector.
Perpendicular Vectors
Magnitude is determined using Pythagorean Theorem
Direction of a reference angle is determined using inverse tangent. The reference angle can then be converted to an angle measured from the +x axis.

16. Adding Vectors that are NOT Parallel or Perpendicular
Adding vectors that are parallel and perpendicular is fairly simple. But, what if vectors are not parallel or perpendicular to one another?
The secret is vector components.
If vectors A and B are added together to produce vector C , then
the x-components of A and B must equal the x-components of C ,
the y-components of A and B must equal the y-components of C ,

17. Steps When Adding Vectors that are NOT Parallel or Perpendicular
Split the vectors into their scalar components. Find the magnitudes of all the components and decide the sign (plus or minus) that is consistent with the direction each component is pointing.
Sum the x-components using ordinary addition as (they are parallel).
Sum the y-components using ordinary addition (they are parallel).
The vector sum of the x-components will always be perpendicular to the vector sum of the y-components. The magnitude of the resultant can be found using Pythagorean theorem.
The direction of the resultant can be found using inverse tangent.
Remember: This will calculate a reference angle.

18. Add the following displacement vectors: 20 m at 45o and 10 m at 30o.
Example 1
+20m
45o
+10m
30o

19. Add the following displacement vectors: 20 m at 45o and 10 m at 30o.
Example 1
Added tip to tail the given vectors would appear as follows
The resultant would extend from the origin to the tip of the last vector.
+10m
30o
+20m
45o

20. Add the following displacement vectors: 20 m at 45o and 10 m at 30o.
Example 1
1. Split the vectors into components
+14.1
+8.66
+5.00
+10m
30o
+20m
45o

21. Example 1 +10m (+5.00) 30o +(+14.1) +19.1 +20m 45o
Add the following displacement vectors: 20 m at 45o and 10 m at 30o.
Example 1
Add the x-components (Ax + Bx = Cx )
Add the y-components (Ay + By = Cy )
+10m
30o
(+5.00)
+(+14.1)
+19.1
+20m
45o
(+14.1) + (+8.66) = +22.8

22. Add the following displacement vectors: 20 m at 45o and 10 m at 30o.
Example 1
The resultant extends from the origin to the tip of the last vector.
This no forms a right triangle, with the resultant as the hypotenuse.
+10m
30o
+20m
45o
+19.1
+22.8

23. Add the following displacement vectors: 20 m at 45o and 10 m at 30o.
Example 1
4. Use Pythagorean theorem to determine the resultants magnitude.
5. Use inverse tangent to determine direction as an angle.
+19.1
+22.8