Transcript Particle Motion - Harding Charter Preparatory High School

Particle Motion
AP Calculus AB
3 Closely related
topics
• Position = s(t) = p(t) = x(t)
• Velocity = v(t)
• Acceleration = a(t)
• x’(t) = v(t)
and
v’(t) = a(t)
Let x(t) be the position of a
particle…
t
• Initially means when _______
= 0.
x(t)
• “At the origin” means _________
= 0.
v(t)
• “At rest” means _______
= 0.
• If the velocity of the particle is positive, then the
right
particle is moving to the _________.
negative
• If the velocity of the particle is ___________,
then the
particle is moving to the left.
Let x(t) be the position
of a particle…
• To find the average velocity over a time interval, divide
position
the change in _____________
by the change in time.
• Instantaneous velocity is the velocity at a single
moment in time.
• If the acceleration of the particle is positive then the
velocity
_______________
is increasing.
negative
• If the acceleration of the particle is _____________,
then the velocity is decreasing.
Let x(t) be the position
of a particle…
• In order for a particle *TO* change directions, the
velocity
___________
must change signs.
velocity
• Speed is the absolute value of _______________.
• If the velocity and acceleration have the same sign, the
increasing
speed is _______________.
(speeding up)
opposite
• If the velocity and acceleration are ______________
in
sign (one positive and the other negative), then the
speed is decreasing. (slowing down)
3 ways to use an
integral
ò•
indefinite
integral. It will give you
v(t)dt is an ______________
an expression for position
_____________ at time t. Don’t forget
“c” value of which can be
that you will have a _____
determined if you know a position value at a particular
time.
3 ways to use an
integral
definite
integral and so the answer will be a
ò• t v(t)dt is a ________
constant The number represents the change in
_______.
position
_________
over the time interval. By the Fundamental
Theorem of Calculus , since v(t) = x’(t), the integral
will yield x(t2) – x(t1). This is known as displacement.
negative
The answer can be positive or __________
depending
right
upon if the particle lands to the ________
or left of its
original starting position.
t2
1
3 ways to use an
integral
definite
integral and so the
ò•t | v(t) | dt is also a ___________
answer will be a number. The number represents the
total distance
________________
traveled by the particle over the
time interval. The answer should always be
positive
___________.
t2
1
Example 1 page 62
• The data in the table below gives selected values for the
velocity, in meters per minute, of a particle moving
along the x-axis. The velocity is a differentiable
function of time t.
t
0
2
5
6
8
12
v(t)
-3
2
3
5
7
5
Example 1
• 1) At t = 0, is the particle moving to the right or to the
left?
• The particle is moving to the left because the velocity at
time = 0 is negative.
• 2) Is there a time during the time interval 0 < t < 12
minutes, when the particle is at rest? Explain your
answer.
• Yes between time 0 and time 2. By IVT, since v(0) < 0 <
v(2) then there must be some time t where 0 < t < 2
where the velocity is equal to zero.
Example 1
• Use data from the table to find an approximation for
v’(10) and explain the meaning of v’(10) in terms of
the motion of the particle. Show the computations that
lead to your answer and indicate the units of measure.
v(12) - v(8)
1
v'(10) =
= - m/ min 2
12 - 8
2
• v’(10) is the acceleration of the object at 10 minutes.
Example 1
• Let a(t) denote the acceleration of the particle at time t.
Is there a guaranteed to be a time t = c in the interval
0 < t < 12 such that a(c) = 0? Justify your answer.
• Between time 0 and 2, there will be a slope of positive
5/2.
• Between time 8 and 12, there will be a slope of negative
1/2.
• Therefore by IVT, there exists some time c between 0 and
12 where a(t) = 0.
Example 2
• 1) At t =4 seconds, is the particle moving to the left or to the
right? Explain.
• Right because the velocity is positive.
• 2) Over what time interval is the particle moving to the left.
Explain.
• Between the time interval (5,9) seconds because the velocity is
negative
• 3) At t=4 seconds, is the acceleration of the particle positive
or negative? Explain.
• Negative because the slope of the graph is negative (v’(t) < 0)
Example 2
• 4) What is the average acceleration of the particle over
the interval [2,4]. Show the computations that lead to
your answer and indicate units of measure.
• Find by using the slope formula with your velocity
function:
v(4) - v(2) 6 - 9
3
2
=
= - feet / second
4-2
2
2
• 5) Is there guaranteed to be a time t in the interval [2,4]
such that v’(t) = -3/2? Justify.
• No because v(t) is not differentiable over the closed
interval.
Example 2
• 6) At what time t in the given interval is the particle
farthest to the right? Explain.
• At time =5. The particle moves to the right during the
time interval (0,5) and then moves left therefore at 5
seconds the particle is the furthest to the right.
• Try Example 3 on your own. Explanations should be
in the form of complete sentences.
Answers:
• Yours should be in the form of complete sentences.
• 1) v(t) = 3t2 – 12t +9 so v(0) = 9 and so right.
• 2) a(t) = 6t – 12 so a(1) = 6 -12 = -6. So v(t) is
decreasing.
• 3) v(t) = 3 (t2 - 4t +3) = 3(t - 3)(t - 1) so v(t) = 0 when t
is equal to 1 or 3 seconds.
+
1
Moving left between time (1,3) seconds.
+
3
• 4) Total distance =
ò
5
0
|3t 2 -12t +9 | dt = 28
Have page 64 out
• I will come by your desk to see if you did it.
Answers to page 64
• A) v(0) > 0 so right
• F) (2,6)
• B) t = 9 and 17
• G) 52.5 feet
• C) t = 9
• H) 16.5 feet
• D) t = 15
• I) t = 9
• E) 3/2 ft/s²
• J) -2 feet
Page 65
• Work out the problem on page 65 with one other
person. I will call on pairs for explanations.
• Feel free to continue working through this packet if
you get done. The packet will be due in entirety on
Tuesday!
Answers to page 65
• A) v(t) = 3t²-12t+9
• F) 20 m
• B) v(2) = -3m/s
v(4) = 9m/s
• G)
• C) v(t) = 0 when t =1,3
• D) (0,1)u(3,infinity)
• E)  

(0,1) (1,3) (3,infinity)
5
|𝑣(𝑡)|
0
≈ 27.999𝑚
• H) a(4) = 12m/s²
• I) Graphs of equations
• J) Speeding up
(1, 2)u(3, infinity)
Slowing Down
(0,1)u(2,3)
Homework
• Finish # 1 and 2 on page 66
• You may cross out those two multiple choice questions
#3 and 4.
Thursday
• Have #1 and 2 out for me to check.
Thursday
• Today in class you are getting time to work on this
packet on your own, with a partner, or with a small
group.
• Tomorrow we will go over the multiple choice
questions.
• So please have a calculator.
• Remember – the whole thing is due completed on
Tuesday!
1997 Ab 87 C
• At time t > 0, the
acceleration of a particle
moving on the x-axis is
a(t) = t +sin(t). At t = 0, the
velocity of the particle is -2.
For what value of t will the
velocity of the particle be
zero?
• A) 1.02
• B) 1.48
• C) 1.85
• D) 2.81
• E) 3.14
1997 Ab 87 C
• At time t > 0, the
acceleration of a particle
moving on the x-axis is
a(t) = t +sin(t). At t = 0, the
velocity of the particle is -2.
For what value of t will the
velocity of the particle be
zero?
• A) 1.02
• B) 1.48
• C) 1.85
• D) 2.81
• E) 3.14
1998 Ab 14 NC
• A particle moves along the
x-axis so that its position at
time t is given by
x(t) = t² - 6t + 5. For what
value of t is the velocity of
the particle zero?
• A) 1
• B) 2
• C) 3
• D) 4
• E) 5
1998 Ab 14 NC
• A particle moves along the
x-axis so that its position at
time t is given by
x(t) = t² - 6t + 5. For what
value of t is the velocity of
the particle zero?
• A) 1
• B) 2
• C) 3
• D) 4
• E) 5
1998 Ab 24 NC
• The maximum acceleration
attained on the interval 0 < t
< 3 by the particle whose
velocity is given by
v(t) = t³ - 3t² + 12t + 4 is
• A) 9
• B) 12
• C) 14
• D) 21
• E) 40
1998 Ab 24 NC
• The maximum acceleration
attained on the interval 0 < t
< 3 by the particle whose
velocity is given by
v(t) = t³ - 3t² + 12t + 4 is
• A) 9
• B) 12
• C) 14
• D) 21
• E) 40
2003 ab 25 NC
• A particle moves along the
x-axis so that at a time t > 0
its position is given by
x(t) = 2t³ - 21t² + 72t – 53.
At what time t is the particle
at rest?
• A) t = 1 only
• B) t = 3 only
• C) t = 7/2 only
• D) t = 3 and t = 7/2
• E) t = 3 and t = 4
2003 ab 25 NC
• A particle moves along the
x-axis so that at a time t > 0
its position is given by
x(t) = 2t³ - 21t² + 72t – 53.
At what time t is the particle
at rest?
• A) t = 1 only
• B) t = 3 only
• C) t = 7/2 only
• D) t = 3 and t = 7/2
• E) t = 3 and t = 4
2003 ab 76 C
• A particle moves along the
x-axis so that at any time t >
0, its velocity is given by
v(t) = 3 + 4.1cos(0.9t). What
is the acceleration of the
particle at time t =4?
• A) -2.016
• B) -0.677
• C) 1.633
• D) 1.814
• E) 2.978
2003 ab 76 C
• A particle moves along the
x-axis so that at any time t >
0, its velocity is given by
v(t) = 3 + 4.1cos(0.9t). What
is the acceleration of the
particle at time t =4?
• A) -2.016
• B) -0.677
• C) 1.633
• D) 1.814
• E) 2.978
2003 ab 83 C
• The velocity, in ft/sec, of a
particle moving along the xaxis is given by the function
v(t) = 𝑒 𝑡 + 𝑡𝑒 𝑡 . What is the
average velocity of the
particle from t = 0 to time t
= 3?
• A) 20.086 ft/sec
• B) 26.447 ft/sec
• C) 32.809 ft/sec
• D) 40.671 ft/sec
• E) 79.342 ft/sec
2003 ab 83 C
• The velocity, in ft/sec, of a
particle moving along the xaxis is given by the function
v(t) = 𝑒 𝑡 + 𝑡𝑒 𝑡 . What is the
average velocity of the
particle from t = 0 to time t
= 3?
• A) 20.086 ft/sec
• B) 26.447 ft/sec
• C) 32.809 ft/sec
• D) 40.671 ft/sec
• E) 79.342 ft/sec
2003 ab 91 C
• A particle is moving along
the x-axis so that at any time
t > 0, its acceleration is
given by a(t) = ln(1+2𝑡 ). If
the velocity of the particle is
2 at time t= 1, then the
velocity of the particle at
time t = 2 is
• A) 0.462
• B) 1.609
• C) 2.555
• D) 2.886
• E) 3.346
2003 ab 91 C
• A particle is moving along
the x-axis so that at any time
t > 0, its acceleration is
given by a(t) = ln(1+2𝑡 ). If
the velocity of the particle is
2 at time t= 1, then the
velocity of the particle at
time t = 2 is
• A) 0.462
• B) 1.609
• C) 2.555
• D) 2.886
• E) 3.346
Back of last page
• A) Finding the definite integral without an equation? What does it
mean? Do we know geometry that can get us there? What does
the definite integral of velocity give us? What units are
appropriate?
• B) Why would a derivative not exist? What does v’(t) represent?
What units are appropriate?
• C) Write the piecewise function for a(t) – something that we have
only really done one time in this class.
• What should we be thinking about? Can we find the slope of the
graph during certain time intervals? Where would the derivative
not exist?
• D) Average rate of change: slope! Does MVT apply? Why or why
not? If so, what will it guarantee?