Describing Motion. the study of motion motion is a change in position two branches Mechanics.

Describing Motion

Describing Motion

• the study of motion• motion is a change in

position• two branches

MechanicsMechanics

• describes how objects move

KinematicsKinematics

DynamicsDynamics• explains the causes of

motion

• Chapter 3 is about one-dimensional motion, as on a number line

Mathematical Representations of Motion—The Basics

Mathematical Representations of Motion—The Basics

• Origin: A reference point• to the left—negative• to the right—positive

Mathematical Representations of Motion—The Basics

Mathematical Representations of Motion—The Basics

• When motion is vertical:• up—positive• down—negative

Mathematical Representations of Motion—The Basics

Mathematical Representations of Motion—The Basics

• An object has moved if at one time its position is x1 and at another time its position is x2.

Mathematical Representations of Motion—The Basics

Mathematical Representations of Motion—The Basics

• An object’s position at a time can be represented by an ordered pair: (t1, x1) or (t2, x2)

Mathematical Representations of Motion—The Basics

Mathematical Representations of Motion—The Basics

DisplacementDisplacement• the change in position

between two distinct points

• often different from the distance traveled

Scalars and VectorsScalars and Vectors

• a scalar contains just one piece of information

• a vector contains two: magnitude and direction

• vectors are represented in bold: d, v, etc.

Scalars and VectorsScalars and Vectors

• for vectors in one-dimensional motion, subscripts may be used, such as dx

• this will represent a change in position

What do we know about the family’s travels?

What do we know about the family’s travels?

a. displacement = 2 km northSince displacement is a

vector, a direction must be indicated.

Example 3-1Example 3-1

What do we know about the family’s travels?

What do we know about the family’s travels?

b. the car has traveled 10 kmSince distance is a scalar, no

direction needs to be indicated.

Example 3-1Example 3-1

What do we know about the family’s travels?

What do we know about the family’s travels?

c. the displacement is zero, since its final and initial positions are the same

When d = 0, no direction is necessary.

Example 3-1Example 3-1

What do we know about the family’s travels?

What do we know about the family’s travels?

d. the car has traveled 20 km

Example 3-1Example 3-1

• plots ordered pairs of data in a simple form

Position-time GraphPosition-time Graph

• allows the calculation of:• displacement• average speed

Position-time GraphPosition-time Graph

• to calculate:

Average SpeedAverage Speed

v =|x2 - x1|

t2 - t1

=s

Δt=

|Δx|Δt

• the speed of an object at any one moment

• the slope of the position-time curve at that point

Instantaneous SpeedInstantaneous Speed

The slope is easy to find if the position-time curve is linear, but what if it is a

curve?We can use a tangent line.

Instantaneous SpeedInstantaneous Speed

Can you see why graph (c) is the best estimate for a

tangent line?

Instantaneous SpeedInstantaneous Speed

Be sure to recognize the difference between average

speed and instantaneous speed.

For which one can you get a speeding ticket??

Instantaneous SpeedInstantaneous Speed

VelocityVelocity• includes both speed

and direction• to calculate average

velocity:

v =d

Δt

VelocityVelocity• displacement (d) might

be positive or negative in one-dimensional motion

v =d

Δt

VelocityVelocity• can be calculated from

a position-time graph• can be positive or

negative

v =d

Δt

• allows the calculation of:• acceleration

Velocity-time GraphVelocity-time Graph

AccelerationAcceleration• change in velocity with

respect to time• to calculate average

acceleration:

a =ΔvΔt

AccelerationAcceleration• acceleration is a vector

pointing in the same direction as Δv

a =ΔvΔt

AccelerationAcceleration• average acceleration

can be calculated as the slope of a velocity-time graph

a =ΔvΔt

AccelerationAcceleration• uniformly accelerated

motion involves a constant rate of velocity change

Equations of Motion

Equations of Motion

First Equation of MotionFirst Equation of Motion

• often used if you want to know the final velocity when you know the initial velocity and acceleration

v2x = v1x + axΔt

Determining Displacement Algebraically

Determining Displacement Algebraically

dx = ½(v1x + v2x )Δt

dx = vxΔt

Determining Displacement Geometrically

Determining Displacement Geometrically

• the area “under the curve” of a velocity-time graph is equal to the displacement of the moving object

Second Equation of Motion

Second Equation of Motion

• two common forms:

dx = v1xΔt + ½ax(Δt)²

x2 = x1 + v1xΔt + ½ax(Δt)²

Third Equation of MotionThird Equation of Motion

• two common forms:• two common forms:

dx =v2x² – v1x²

2ax

x2 = x1 +v2x² – v1x²

2ax

Equations of MotionEquations of MotionThese are used to solve most problems involving

straight-line, constant acceleration motion.

Sometimes there will be more than one possible

method.

Free FallFree Fall• an object falls under the

influence of gravity alone with negligible air resistance

• near earth’s surface:

g = gy = -9.81 m/s²

Free FallFree Fall• the equations of motion are

easily adapted by replacing the acceleration with gy:

v2y = v1y + gyΔtFirst Equation of Motion:

Free FallFree FallSecond Equation of Motion:

dy = v1yΔt + ½gy(Δt)²Third Equation of Motion:

2gy

dy =v2y² – v1y²