Chapter 3 Reading Guide

3.1 *What is the difference between a scalar and a vector?

The methods involved in solving physics problems that involve scalars are different than the methods involved in solving problems which involve vectors. You must learn to recognize whether the problem has a scalar nature or a vector nature. In most cases in this book, vector problems require you to work 2 problems simultaneously. One calculation will be in the X direction and the other calculation will be in the Y direction. Often the X and Y results will be combined through the use of right-triangle calculations.

*In what way is the magnitude of a vector equal to the size of the vector?

3.2 Graphical Addition of Vectors

The addition of vector quantities does not follow the ordinary rules of algebra. You can only perform ordinary addition or subtraction if the vectors are parallel to each other. Vectors can be added graphically to obtain an approximate result. Vectors can be added mathematically to obtain an exact result. The sum of two vectors is called the resultant.

What is the approximate addition of a vector of 5 units to the east and a vector that has a length of 7 units in the northwest direction? Use the tail-to-tip method. I don’t use the parallelogram method.

 

 

 

 

 

 

 

Vector addition does posses the property called commutation. That is, you can start the sum process with any of the vectors that need to be added and still obtain the same result. A + B is equal to B + A .

3.3 Vector Subtraction and Multiplication of a Vector by a Scalar

Vector subtraction A – B is accomplished by adding the negative of the B vector. The negative of a vector has the same magnitude but a direction 180 degrees different than the positive version of the vector. On a graph you simply draw the arrow at the other end of the vector.

Scalar multiplication of vectors involves multiplying a vector by a scalar, e.g. 3A. Scalar multiplication affects the magnitude but not the direction of the vector. If A has a magnitude of 5, 3 A has a magnitude of 15.

 

3.4 Adding Vectors by Components

To perform the mathematical addition of vectors we must be able to resolve vectors into their components. This is accomplished by the trigonometric calculations for a right triangle. The vector is always the hypotenuse in the right triangle. The X and Y components are the sides of the right triangle. If the angle is given in standard form (i.e. measured counterclockwise from the positive X axis) then you can always use the relationships below to find the components of the vector. If A is the magnitude of the vector then

*AX = A cos q and AY = A sin q .

Suppose a certain vector has a magnitude of 12 meters and a direction of 37 degrees from the positive X axis. What are the values of the components of the vector?

AX Ay

 

If you are given the components of a vector you can reconstruct the vector using the Pythagorean Theorem and the tangent function (plus thinking).

* A = sqrt( AX + AY ) tan q = AY /AX How is q found from this relationship? Is it always the correct angle?

Perform the reconstruction of A using the AX and AY you found above.

Mathematical addition of vectors ( C = A + B ) is performed by 1) finding the components of each vector A and B, 2) adding "like" components e.g. CX = AX + BX , 3) reconstructing C.

Suppose A has a length of 3 meters and an angle of 30 degrees, B has a length of 7 meters and an angle of 160 degrees. Find C = A + B.

 

 

 

 

 

 

When a problem has two dimensions you must work the problem using vector techniques. Make a vector diagram before you attempt to solve the problem.

 

3.5 Projectile Motion Concepts

*Displacement, velocity and acceleration are vectors. When motion occurs in a direction not aligned with either the X or Y axis you must work two problems simultaneously. *It is very important that you keep all X quantities in one equation and all Y quantities in a separate equation. The only quantity which will be common to both equations is the time value.

Your result will have components of each quantity. You will need to reconstruct the vector to determine the magnitude and angle of the displacement or velocity or acceleration.

TRUE or FALSE It is possible for an object which has constant speed to have a non-zero value for its acceleration.

*Projectile motion refers to motion which takes place near the surface of a large mass (e.g. the earth). Perhaps your instructor will throw a soft object at a student who is sleeping at this early hour of the morning. Watch the object carefully as it flies through the air. The object is undergoing projectile motion. In projectile motion the vertical acceleration is 9.81 m/s2 (assuming the object is moving near the earth, not mars etc.)

What is the value of the horizontal acceleration?

What is true about the horizontal velocity?

We will look at some transparencies that illustrate projectile motion.

3.6 Solving Projectile Motion Problems

The acceleration in the x direction is zero when we ignore air resistance. This causes the kinematic equations for the x motion to simplify to distance = rate * time. The y (vertical) equations will contain -9.8 m/s2 for the acceleration value.

*Did anyone have a question on any of the examples worked in the text?

 

3.7 Projectile Motion is Parabolic

PHY161 only

3.8 Relative Velocity

Review the handout that shows a boat crossing a river and calculates the relative velocity of the boat with respect to the shore.

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