Describing Motion Describing Motion Chapter 3 Chapter 3
Jan 03, 2016
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
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.
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.
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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