HCPSS Worthwhile Math Task
Particles in Motion
Content Standard(s)
Goal: The student will demonstrate the ability to use a problem-solving approach in solving a
variety of problems involving applications of the derivative.
Objectives: The student will be able to interpret the derivative as a rate of change in a variety of
applied contexts, including velocity, speed, and acceleration.
MP1:
MP2:
MP4:
MP5:
MP6:
MP7:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
AP Calculus AB, Unit IX
The Task
A particle moves along a straight line for 0 ≤ t ≤ 2π, where t is measured in seconds. The
velocity of the particle is given by v(t )  8 sin 2 ( x   )  8 cos( x   )  8 . The initial position
2
2
s(0) =2.
1. Describe the motion of the particle over the given time period. Show the mathematics that
leads to your conclusion.
2. Describe the position of the particle over the given time period. Show the mathematics that
lead to your conclusion.
3. Is the speed up or slowing down at t  3 ? Explain your reasoning.
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4. What is the particles displacement and total distance traveled over the given time period?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Facilitator Notes
1. Have students complete the description independently with the use of a calculator. Then
have students get into groups of three or four, and share their description. Students
should be adding to their description during the group share. (Look for evidence of MP1,
MP2, and MP3.)
2. Students may interpret the graph or use the function to get the solutions. (Look for
evidence of MP5 and MP7.)
3. Be sure students address velocity and acceleration when describing the motion of the
particle. They may make a sign chart, but be sure they interpret what the sign chart
represents.
Follow-Up Questions
1. Will the particle ever have a negative position? Justify your answer.
2. Find all exact times where the particle changes direction? Show all work.
3. What is the maximum acceleration during the given time period?
Solutions
1. v(t)=0 at t   t   , and t  2 . a(t)=0 at t   t   , t  5 and 3
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6
2
6
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When the velocity is negative, the particle is moving left, and when the velocity is positive, the
particle is moving right. When the sign of the acceleration matches the sign of the velocity the
particle is speeding up, and when they do not match, the particle is slowing down. Students can
find this information graphically or by solving the velocity and acceleration functions equal to
zero.
2. The particle changes direction where v(t) = 0, ( v(t)=0 at t   t   , and t  2 . ) The
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particle is at 2 when t = 0, it is at -1.434 when t = π, and at 27.133 when t = 2π. These solutions
t
are found by finding  0 v (t ) dt  2 for each of the places where the velocity function crosses
the x-axis.
3. At t  3 the velocity is -1.657, and the acceleration is -2.343, since both are negative, the
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speed is increasing in a negative direction.
4. What is the particles displacement and total distance traveled over the given time period?
2
2
Displacement =  0 v (t ) dt =25.133
Total Distance =  0 v (t ) dt =32
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Follow Up Questions:
1. Yes, at t = π, the position is -1.434.
2. The particle changes direction where v(t) = 0, ( v(t)=0 at t   t   , and t  2 . )
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3. To maximize acceleration v(t )  0 , and changes from positive to negative. So the options
for maximums are at t  1.003, t  3.776, and t  7.286 . Substituting those values back into
v(t )  a (t ) to see which gives the highest value. The maximum acceleration is 14.081 when
t=3.776.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.