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


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



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



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



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



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?


a. displacement = 2 km northSince displacement is a

vector, a direction must be indicated.

Example 3-1Example 3-1



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

direction needs to be indicated.




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

When d = 0, no direction is necessary.




d. the car has traveled 20 km


• 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.


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

tangent line?


Be sure to recognize the difference between average

speed and instantaneous speed.

For which one can you get a speeding ticket??


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

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

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