1
Lesson Plan #79
Date: Tuesday April 21st, 2015
Class: PreCalculus
Topic: Particle Motion Questions
Aim: How do solve particle motion questions?
Objectives:
1) Students will be able to solve particle motion questions.
HW# 79:
1)The position of a particle moving along a straight line is given by s  t 3  6t 2  12t  8
A) Find the interval(s) for which s is increasing
B) Find the minimum value of the speed
C) Find the interval(s) for which acceleration is positive
D) Find the interval(s) for which the speed of the particle is decreasing
Do Now:
If the position of a particle at a time t is given by the equation x(t )  t 3  11t 2  24t , find the velocity and acceleration
of the particle at time t  5
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Online Interactive Activity : Let’s go to
http://www.geogebra.org/en/upload/files/english/lewws/ex14cq12r1.html
to help us visualize motion of a particle along a straight line.
Particle Motion Along A Line:
If a particle moves along a line according to the law s  f (t ) , where
s represents the position of the particle P on the line at
ds
dv
d 2s
time t , then the velocity v of P at time t is given by
and its acceleration a by
or by
. The speed of the particle v ,
dt
dt
ds 2
is the magnitude of v .
NOTES:
1) If v  0 , then P is moving to the right and its distance
s is decreasing.
3) If
If
s is increasing. If v  0 , then P is moving to the left and its distance
a and v are both positive or both negative, then the speed of P is increasing or P is accelerating.
a and v are have opposite signs, then the speed of P is decreasing or P is decelerating.
4) If s is a continuous function of t , then P reverses direction whenever
velocity does not necessarily imply a change in direction.
v is zero and a is different from zero. Note that zero
2
Example #1:
If the position of a particle is given by x(t )  t  12t  36t  18 , where
A) Find the point at which the particle changes direction.
3
2
t  0,
B) Find the interval during which the particle is slowing down.
Example #2:
If its position function of a particle is
A) the eighth and tenth seconds
B) between
x(t )  t 2  6t ,how far does a particle travel between
t  2 and t  4
Example #3:
Given the position function
x(t )  t 4  8t 2 , find the distance the particle travels from t  0 to t  4
Example #4:
Given the position function
x(t )  2t 3  21t 2  60t  3 , for t  0 , find when the particle’s speed is increasing
3
Sample Test Questions:
4
Sample Test Questions:
1)
For this question, focus on the
velocity increasing. Velocity is
the first derivative. To see if it
is increasing, you have to check
the 2nd derivative and see if it is
where it is positive, or visually
where it is concave up.
2)
For this question, focus on
which way is the particle
moving at t = 0. That should
eliminate some choices. Then
look at how fast it is moving at
t = 1 and that should eliminate
all the other choices
5
3)
Do only parts C and D for this
question. For C recall speed is
decreasing if velocity and
acceleration have different signs.
For D, since this is the graph of
velocity and they want to see
where acceleration is negative
see where the velocity is
decreasing.
4)