1 3.4 velocity & other rates of change annecy, french alps
TRANSCRIPT
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13.4 Velocity & Other Rates of Change
Annecy, French Alps
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Slopes of Displacement FunctionsThe slope of a function is equal to the rate of change of that function:
dx
dySlope = rate of change =
Let f(t) be a function that describes how far an object is from its starting point (displacement)
ntdisplaceme)( tfs
dt
ds)(' tf
t
tfttft
0
lim
Time
Distance
VelocityousInstantane
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Velocity versus Speed• In Calculus velocity and speed are not the same thing.
t
tfttf
dt
dstv
t
0
lim)(
Instantaneous Velocity:
Speed:
dt
dstv )(Speed
Speed does not include direction.
Velocity includes direction.
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Velocity in Graphs
M = Average Speed
)(tvdt
ds
S(t)
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Freefall Constants• On the planet earth, the gravitational force is constant.
In feet:
G= 32 feet per second per second
2s
ft32g
In meters
G= 9.8 meters per second per second
2s
m8.9g
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Freefall on Earth• The formulas for freefall are shown below. They are used to
calculate the position (s) of any object at any given time (t).
In feet:
00232
2
1)( stvtts
In meters
0028.9
2
1)( stvtts
00216)( stvtts 00
29.4)( stvtts
Initial Velocity
Initial Position
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Freefall and Derivatives• Using the formula for freefall, formulas for
velocity and acceleration can be quickly computed.
00216)( stvtts
032 vtdt
ds velocity)( tv
322
2
dt
sdvelocityinchange
dt
dv
onaccelerati)( ta
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Freefall and Derivatives• A penny is dropped off the top of the Eiffel Tower (972
ft). When it hits the ground, how fast is it traveling?
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Freefall and Derivatives• A penny is thrown straight up off the top of the Eiffel
Tower (972 ft). It initial speed is 64 feet per second. When it hits the ground, how fast is it traveling?
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Freefall and Derivatives• A penny is thrown straight down off the top of the Eiffel
Tower (972 ft). It initial speed is 64 feet per second. When it hits the ground, how fast is it traveling?
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A person on ground level throws an object upwards such that the height, h, in feet at time t, in seconds, is given by h(t) = -16t2 + 68t.
a. Find the velocity and acceleration.b. Determine the displacement over the interval
[1, 4].c. Determine the average velocity over the
interval [1, 4].d. Determine when the object hits the ground.e. Find the velocity and acceleration of the object
at the time it hits the ground.
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Particle Motion Problems• Atomic particle are subject to many different forces besides
gravity. Therefore, the motion of a particle can take a multitude of forms. To limit the possibilities, we will only deal with particles that travel forward & backward on a line.
Example:
A particle is moving along a line. The position of the particle can be described by the equation:
08)( 24 tttts
Describe the motion of the particle.
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Particle Motion ProblemsA particle is moving along a line. The position of the particle can be described by the equation:
08)( 24 ttttsDescribe the motion of the particle.
Here is a graph of s(t):
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Particle Motion ProblemsA particle is moving along a line. The position of the particle can be described by the equation:
08)( 24 tttts
tttv 164)( 3
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Particle Motion ProblemsA particle is moving along a line. The position of the particle can be described by the equation:
08)( 24 tttts
tttv 164)( 3 1612)( 2 tta
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Particle Motion ProblemsA graph of a particle velocity is shown.
1.) What does the graph of position look like?
2.) What does the graph of acceleration look like?
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Particle Motion ProblemsA graph of a particle velocity is shown.
1.) What does the graph of position look like?
2.) What does the graph of acceleration look like?
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Particle Motion ProblemsA graph of a particle acceleration is shown.
1.) What does the graph of velocity look like?
2.) What does the graph of position look like?
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A particle moves along the x-axis and its positionat time t is given by s(t) = t3 – 12t2 + 27t, where t is measured in seconds and s in feet.
a. Find the velocity and acceleration.b. Determine the velocity and acceleration after 5
seconds.c. Determine when the particle is at rest.d. Determine when the particle is moving forward.e. Find the displacement of the particle during the first 6
seconds.f. Determine the velocity of the particle when there is no
acceleration.
0t
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A particle moves along the x-axis and its positionat time t is given by , where t is measured in seconds and s in feet.
a. Find the velocity and acceleration.b. Determine the velocity and acceleration after 5
seconds.c. Determine when the particle is at rest.d. Determine when the particle is moving forward.e. Find the displacement of the particle during the first 6
seconds.f. Determine the acceleration of the particle when
velocity is at a minimum.
2)2)(3()( ttts 0t