Notes

Vectors and Scalars	Multiplication of Vectors by a Scalar	Projectile Motion
Addition of Vectors: Graphically	Adding Vectors by Components
Subtraction of Vectors	General Kinematic Equations for Constant a

 

Vectors and Scalars

¢ A vector is a quantity that has direction and magnitude

¢ A scalar is a quantity that can be expressed in terms of its magnitude only

Addition of Vectors: Graphically

First Method

 ¢ Draw the vectors to scale on graph paper - appropriate for the problem, and in accordance with the following

         Ø The beginning of the vector is referred to as the tail, the end of the vector is referred to as the tip

         Ø Draw each new vector tail to tail of the previous one. The order is not important; it does not affect the result

 ¢ The arrow drawn from the tail of the first vector to the tip of the last vector represents the sum of the vectors.

         The length is the magnitude of the sum, the direction is as indicated. 

      

Second Method: Parallelogram method - equivalent to the above

Subtraction of Vectors

So, the approach discussed above, tail to tip method, works for subtraction with the above understanding

Multiplication of Vectors by a Scalar

This gives a vector whose magnitude is 1.3 times greater and in the same direction

Adding Vectors by Components

 ¢ Graphics methods are often not very precise and not useful for operations in 3 dimensions

 ¢ A vector can be resolved into and expressed as 2 components, one along the x axis and one along the y axis

 ¢ Trig relationships

 

We use the trig functions as discussed here. Assume a right triangle with base = a, height = b, and hypotenuse = c.

Assume the angle between the base and the hypotenuse is referred to as theta (q). With this configuration

 

sin theta = b/c                                                   

cos theta = a/c

tan theta = b/a

 ¢ Pythagorean Theorem

 

 

 

in the above, the negative option is not considered

 

 ¢ A useful trigonometric identify that follows from the Pythagorean Theorem

 

 

 ¢ Inverse Tangent

If the tangent of an angle theta is b/a, then the inverse tangent (arc tangent as usually called) is the angle whose tangent is b/a.

It is written theta = arctan(b/a)

General Kinematic Equations for Constant Acceleration

x component (horizontal)	y component (vertical)	Comments
vx = v0x + axt	vy = vy0 + ayt	same except for direction
x = x0 + vx0t + (1/2)axt2	y = y0 + vy0t + (1/2)ayt2	same except for direction
v2x = v2x0 + 2ax(x - x0)	v2y = v2y0 + 2ay(y - y0)	same except for direction

Projectile Motion

Kinematic Equations for Projectile Motion

Note that these equations are specific cases of the general equations with stated assumptions

 

Assuming y is positive upwards, a_x = 0, a_y = -g

Horizontal Motion: ax = 0, vx = constant	Vertical Motion: ay = -g	Comments
vx = vx0	vy = vyo - gt	ax = 0, ay ≠ 0
x = x0 + vx0t vx = vx0 (same as 1st one-reason not listed in text)	y = y0 + vy0t - (1/2)gt2 v2y = v2y0 - 2g(y - y0)	ax = 0, ay ≠ 0, ay = -g ax = 0, square root of both sides

NOTE: Equations are provided for all review exercises

m into x and y components

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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