Describing MotionDescribing Motion

Chapter 3Chapter 3

What is a motion diagram?What is a motion diagram?

A Motion diagram is a useful tool to study A Motion diagram is a useful tool to study the relative motion of objects.the relative motion of objects.

From motion diagrams, it is possible to From motion diagrams, it is possible to observe an object under:observe an object under: Constant velocityConstant velocity Accelerating positivelyAccelerating positively Accelerating negativelyAccelerating negatively Or StationaryOr Stationary

Motion DiagramsMotion DiagramsConstant Speed:

Negative Acceleration:

Positive Acceleration:

The Particle ModelThe Particle Model

To simplify motion diagrams, we can To simplify motion diagrams, we can concentrate all the motion through a single concentrate all the motion through a single point at or near the center of gravity.point at or near the center of gravity.

The Particle ModelThe Particle ModelConstant Speed:

Negative Acceleration:

Positive Acceleration:

The Particle ModelThe Particle ModelConstant Speed:

Negative Acceleration:

Positive Acceleration:

Determining MotionDetermining Motion

An object’s motion can be determined if its An object’s motion can be determined if its initial and subsequent positions are initial and subsequent positions are identified relative to time.identified relative to time.

Initial Time = ti

Initial Position = di

Initial Velocity = vi

Final Time = tf

Final Position = df

Final Velocity = vf

Average VelocityAverage Velocity

The average velocity is the ratio of displacement The average velocity is the ratio of displacement and time as follows:and time as follows:

Where:Where: d = the displacement vector t = change in time ti and di represent the starting

position tf and df represent the final

position

Average velocity does not tell you how the velocity Average velocity does not tell you how the velocity varied during the time interval between the points, dvaried during the time interval between the points, dii and dand dff..

vv = = = =ddtt

ddff - d - dii

ttff - t - tii

(1)

Graphical Representation of Graphical Representation of VelocityVelocity

A graph of an object A graph of an object moving at constant moving at constant velocity will consist of velocity will consist of a straight line.a straight line.

The slope of this line The slope of this line will equal the average will equal the average velocity of the object.velocity of the object.

0

2

4

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18

0 2 4 6 8 10

Time (s)

Po

siti

on

(m

)

Average AccelerationAverage Acceleration

An object in motion with changing velocity is An object in motion with changing velocity is under acceleration under acceleration

Acceleration is the rate of change of velocity as Acceleration is the rate of change of velocity as follows:follows:

As with average velocity, the average As with average velocity, the average acceleration does not tell you how it varied acceleration does not tell you how it varied during the time interval tduring the time interval t ii to t to tff..

aa = = = =vvtt

vvff - v - vii

ttff - t - tii

(2)(2)

Graphical Representation of Graphical Representation of Average AccelerationAverage Acceleration

A graph of an object A graph of an object moving at constant moving at constant acceleration will consist acceleration will consist of a straight line.of a straight line.

The slope of this line will The slope of this line will equal the average equal the average acceleration of the object.acceleration of the object.

The average between the The average between the initial and final values for initial and final values for velocity will equal the velocity will equal the average.average.

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0 1 2 3 4 5

Time (s)

Vel

oci

ty (

m/s

)

vi

vf

vavg

Finding Final Velocity Under Finding Final Velocity Under Uniform AccelerationUniform Acceleration

To find the final velocity when acceleration is To find the final velocity when acceleration is uniform, all that is needed is the initial velocity, uniform, all that is needed is the initial velocity, acceleration and time.acceleration and time.

By rearranging 2 to isolate vBy rearranging 2 to isolate v ff, we obtain:, we obtain:

An alternative method for calculating the final An alternative method for calculating the final velocity is:velocity is:

vf = vi + att (3)

vf2 = vi

2 + 2add (4)

Average Velocity during Uniform Average Velocity during Uniform AccelerationAcceleration

For an object moving at constant acceleration, For an object moving at constant acceleration, the average velocity is equal to the average of the average velocity is equal to the average of the initial plus final velocities.the initial plus final velocities.

vvavgavg = = vvii + v + vff

22(5)

Finding Displacement Under Finding Displacement Under Uniform AccelerationUniform Acceleration

When acceleration is uniform, the displacement depends When acceleration is uniform, the displacement depends on the objects acceleration, initial velocity and time.on the objects acceleration, initial velocity and time.

To find the displacement of an object during uniform To find the displacement of an object during uniform acceleration, substitute 1 into 5 for vacceleration, substitute 1 into 5 for vavgavg..

vvavgavg = = d/d/t t (1)(1)

vvavgavg = = vvii + v + vff

22(5)(5)

d = (vd = (vii + v + vff) ) tt (6) (6)12

d/d/t = t = vvii + v + vff

22

Finding Displacement Under Finding Displacement Under Uniform AccelerationUniform Acceleration

An alternative expression for (6) can be obtained by An alternative expression for (6) can be obtained by substituting 3 into 6:substituting 3 into 6:

d = vd = vi i t + a(t + a(t)t)22 (7) (7)12

d = (vd = (vii + v + vff) ) tt (6) (6)12

d = (vd = (vii + v + vii + a + at)t)tt12

d = [2vd = [2vi i t + a(t + a(t)t)22]]12

vvff = v = vii + a + att (3) (3)

Formulas for Motion of ObjectsFormulas for Motion of Objects

Equations to use when Equations to use when an accelerating object an accelerating object has an initial velocity.has an initial velocity.

Form to use when Form to use when accelerating object accelerating object starts from rest (vstarts from rest (v ii = 0). = 0).

d = ½ (vd = ½ (vi i + v+ vff) ) tt d = ½ vd = ½ vff tt

vvff = v = vii + a + att vvff = a = att

d = vd = vii t + ½ a(t + ½ a(t)t)22 d = ½ a(d = ½ a(t)t)22

vf2 = vi

2 + 2add vf2 = 2add

Formulas for Motion of ObjectsFormulas for Motion of Objectsassuming d is displacement from origin and time starts at 0.assuming d is displacement from origin and time starts at 0.

Equations to use when Equations to use when an accelerating object an accelerating object has an initial velocity.has an initial velocity.

Form to use when Form to use when accelerating object accelerating object starts from rest (vstarts from rest (v ii = 0). = 0).

d = ½ (vd = ½ (vi i + v+ vff) t = v) t = vaveavett d = ½ vd = ½ vff t t

vvff = v = vii + at + at vvff = at = at

d = vd = vii t + ½ a(t) t + ½ a(t)22 d = ½ a(t)d = ½ a(t)22

vvff22 = v = vii

2 2 + 2ad+ 2ad vvff22 = 2ad = 2ad