ChemPhys 173/273

Unit 7: Vectors and Projectiles

Problem Set A

Overview:

Problem Set A targets your ability to perform vector operation such as vector addition and vector resolution in order to determine the magnitude and direction of an unknown vector or vector component. 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.

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. The use of this convention can be reviewed at The Physics Classroom.

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. These basic principles of right triangle trigonometry can be reviewed at The Physics Classroom.

Resolving an Angled Vector into Right Angle Components

A 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. The method of vector resolution can be reviewed at The Physics Classroom.

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 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:

• give attention to the units of stated quantities and peform proper conversions.
• 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.
• employ the habits of a good problem-solver.

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.

Vectors and Direction | Vector Addition | Resultants

View Sample Problem Set.

 Problem Description Audio Link 1 Analysis of a three-legged walk through a maze; the magnitude of the third displacement (leg) must be determined. 2 Referring to the previous problem; determine the direction of the third displacement vector. 3 Determine the magnitude of the resulting displacement of a football based on the addition of three displacement vectors; two vectors are at right angles. 4 Determine the magnitude of the resulting displacement of a shopping cart based on the addition of three displacement vectors; two vectors are at right angles. 5 Referring to the previous problem; determine the direction of the resultant vector. 6 Add two eastward displacement vectors to determine the net displacement. 7 Add an eastward and a westward displacement vector to determine the net displacement. 8 Determine the magnitude of the resulting displacement of a roller coaster car based on the addition of three displacement vectors. 9 Determine the magnitude of the resulting displacement of a dog based on the addition of three displacement vectors; two vectors are at right angles. 10 A jogger makes a two-leg displacement; the magnitude of the second leg of the trip must be determined if given the net displacement. 11 Referring to the previous problem; determine the direction of the second leg of the trip. 12 Resolution of a vector to determine a horizontal component. 13 Referring to the previous problem; determine a vertical component. 14 Resolution of a vector to determine a vertical component. 15 Addition of five right angle vectors (in the Vector Addition lab) in order to determine the magnitude of the resultant.

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15

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