**Unit 7: Vectors and
Projectiles**

**Problem Set B**

**Overview:**

Problem Set B targets your ability to use the analyze physical situations involving the addition of displacement vectors and to analyze relative velocity situations. Right angle trigonometry and vector addition principles will be used to analyze physical situations involving displacement vectors and to interpret and solve word story problems. Understanding of the following concepts will be crucial to your success on this problem set.

**Direction: the Counter-Clockwise from
East Convention**

A vector is a quantity which has magnitude and direction. Most of us are familiar with the map convention for the direction of a vector; on a map, up on the page is usually in the direction of North and to the right on the page is usually in the direction of east. In Physics, we utilize the map convention to express the direction of a vector. When a vector is neither north or south or east or west, an additional convention must be used. One convention commonly used for expressing the direction of vectors is the counter-clockwise from east convention (CCW). The direction of a vector is represented as the counter-clockwise angle of rotation which the vector makes with due East.

**Vector Addition**

Often times a motion involves several segments or legs. For
instance, a person in a maze makes several individual displacements
in order to finish some distance out of place from the starting
position. Such individual displacement vectors can be added using a
head-to-tail method of vector addition. If adding vector B to vector
A, then vector A should first be drawn; then vector B should be added
to it by drawing it so that the tail of vector B starts at the
location that the head of vector A ends. The
**resultant vector** is then drawn
from the tail of A (starting point) to the head of B (finishing
point). The resultant is equivalent to the sum of the individual
vectors. In this problem set, you will have to be able to read the
word story problem and sketch an appropriate vector addition diagram.

Vectors which are added at right angles to each other will sum to a resultant vector which is the hypotenuse of a right triangle. The Pythagorean theorem can be used to relate the magnitude of the hypotenuse to the magnitudes of the other two sides of the triangle. The angles within the right triangle can be determined from knowledge of the length of the sides using SOH CAH TOA.

If one of the vectors to be added is not directed due east, west, north or south, then vector resolution can be employed in order to simply the addition process. Any vector which makes an angle with to one of the axes can be projected onto the axes to determine its components. SOH CAH TOA can be used to resolve such a vector and determine the magnitudes of its x- and y- components. By resolving an angled vector into x- and y-components, the components of the vector can be substituted for the actual vector itself and used in solving a vector addition diagram. The resolution of angled vectors into x- and y-components allows a student to determine the magnitude of the sides of the resultant vector by summing up all the east-west and north-south components.

**Relative Velocity
Situations**

Often times an object is moving within a medium which is moving relative to its surroundings. For instance, a plane moves through air which (due to winds) is moving relative to the land below. And a boat moves through water which (due to currents) is moving relative to the land on the shore. In such situations, an observer on land will observe the plane or the boat to move at a different velocity than an observer in theboat or the plane would observe. It's a matter of reference frame. One's perception of a motion is dependent upon one's reference frame - whether the person is in the boat, the plane or on land.

In a relative velocity problem, information is typically stated about the motion of the plane relative to the air (plane velocity ) or the motion of the boat relative to the water (boat velocity). And information about the motion of the air relative to the ground (wind velocity or air velocity) or the motion of the water relative to the shore (water velocity or river velocity ) is typically stated. The problem centers around relating these two components of the plane or boat motion to the resulting velocity. The resulting velocity of the plane or boat relative to the land is simply the vector sum of the plane or boat velocity and the wind or river velocity.

The approach to such problems demands a careful reading (and re-reading) of the problem statement and a careful sketch of the physical situation. Efforts must be made to avoid mis-interpretting the physical situation. Once properly set-up, the algebraic manipulations become relatively simply and straight-forward. The crux of the problem is typically associated with the reading, interpretting and understanding of the problem statement.

**Habits of an Effective
Problem-Solver:**

An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problem-solver ...

- ... reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it (e.g., a vector addition diagram).
- ... identifies the known and unknown quantities in an organized manner, often times recording them on the diagram iteself. They equate given values to the symbols used to represent the corresponding quantity (e.g., A = 3.6 m, West; B = 4.5 m, South; C = ???).
- ... plot a strategy for solving for the unknown quantity; the strategy will typically center around the use of vector addition and/or resolution principles.
- ... identify the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
- ... perform substitutions and algebraic manipulations in order to solve for the unknown quantity.

To be successful on this problem set, you will need to be able to:

- be able to sketch a vector addition diagram.
- be able to use SOH CAH TOA and Pythagorean theorem to relate the angles and sides of a right triangle.
- be able to read a relative velocity problem and appropriately decipher information which is pertinent to the solution.
- relate the distance of travel to the speed of travel and the time of travel: d = v•t .
- employ the habits of a good problem-solver.

**Additional Readings/Study
Aids:**

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Relative Velocity and Riverboat Problems | Independence of Perpendicular Components

View Sample Problem Set.

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