Motion: 1-Dimensional Kinematics

Kinematics is the science of describing the motion of objects using

words, diagrams, numbers, graphs, and equations.

The goal of any study of kinematics is to develop sophisticated mental models which serve to describe (and ultimately, explain) the motion of real-world objects.

We will begin by looking at some of nature’s basic rules—physics

THIS IS HOW WE WILL ACCOMPLISH THIS FEAT!!!!

First: EXPLORE (ignite your interest)

Secondly: CONCEPT DEVELOPMENT (expand understanding)

Lastly: APPLICATION (problem solving, critical thinking!)

Scalars and Vectors

•Scalars are quantities which are fully described by a magnitude (or numerical value) alone.

•Vectors are quantities which are fully described by both a magnitude and a direction.

The mathematical quantities which are used to describe the motion of objects can be divided into two categories. The quantity is either a vector or a scalar. These two categories can be distinguished from one another by their distinct definitions:

Scalars or Vector?Quantitya. 5 m

 

b. 30 m/sec, East

 

c. 5 mi., North

 

d. 20 degrees Celsius

 

e. 256 bytes

 

f. 4000 Calories

scalar

vector

vector

scalars

scalars

scalars

Distance vs. DisplacementDistance and displacement are two quantities which may seem to mean the same thing yet have distinctly different definitions and meanings.

•Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion.

•Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's overall change in position

Test your KnowledgeUse the diagram to determine the resulting displacement and the distance traveled by the skier during these three minutes.

Distance is…..

Displacement is…..

420 meters

140 m, rightward.

Test Your KnowledgeWhat is the coach's resulting displacement and distance of travel?

Coach’s distance is…

Coach’s displacement is

95 yards55 yards, left.

Working with the Coordinate Plane

Conventions for Describing Directions of Vectors

Representing the Magnitude of a Vector

8 + 6 = ?

Vector Addition: The Order Does NOT Matter

The most common method of adding vectors is the graphical method of head-to-tail addition.

As shown in the animation, the order in which two or more vectors are added does not effect the outcome. Adding A + B + C + D + E yields the same result as adding C + B + A + D + E or D + E + A + B + C! The resultant, shown as the green vector, has the same magnitude and direction regardless of the order in which the five individual vectors are added.

Vector Problems Continued

Plane Problem:

The vectors are added together to get the…..resultant

Vector Problems ContinuePlane Problem:

The vectors are added together….to get the resultant

Vector Problems ContinuedPlane Problem:

Now consider a plane traveling with a velocity of 100.0 km/hr, South which encounters a side wind of 25.00 km/hr, West. Now what would the resulting velocity of the plane be?

Since the two vectors to be added - the southward plane velocity and the westward wind velocity - are at right angles to each other,

the Pythagorean theorem can be used.

What is the answer? 103.1 km, SW

Vector Problems ContinuePlane Problem:

Practice Problem:

The Smith family begins a vacation trip by driving 700.0 km west. Then the family drives 600.0 km south, 300.0 km east, and 400.0 km north. Where will the Smiths end up in relation to their starting point? SOLVE GRAPHICALLY!!!

700.0 km

600.0km

300.0 km

400.0 km

How can sine, cosine, and tangent be used in this situation

REMEMBER THIS mnemonic:

“SOH CAH TOA”The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse

The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse.

The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Sine, Cosine, or Tangent functions.

Since the plane velocity and the wind velocity form a right triangle when added together in head-to-tail fashion, the angle between the resultant vector and the southward vector can be determined using either the sine, cosine, or tangent functions.

The tangent function can be used; this is shown below: tan (theta) =

(opposite/adjacent)

tan (theta) = (25/100)

theta = invtan (25/100)

theta = 14.0 degrees

Start with: sin a° = opposite/hypotenuse

sin a° = 18.88/30

sin a° = 0.6293...

Inverse Sine: a° = sin-1(0.6293...)The angle "a" is 39.0°

Ralph is mowing the back yard with a push mower that he pushes downward with a force of 20.0 N at an angle of 30.0 to the horizontal. What are the horizontal and vertical components of the force exerted by Ralph?

PRACTICE PROBLEM

20.0 N

30.0

30.0 DEGREE ANGLE

Fx

Fy

sin 39° = opposite/hypotenuse

sin 39° = d/30Multiply both sides by 30

d = 30 x Sin 39 The depth "d" is 18.88 m