We first learned about derivatives and then explored their
practical applications with motion along a line (position,
velocity, and acceleration).
Now…
It is time to see how our knowledge of the calculus of
parametric equations can be used to describe motion along a
curve (any direction we want!!)
To do this, we will use VECTORS!!!
Vectors – A Super Brief Over View and
An Intro to the Calculus of Them
Vector – A directed line segment that represents any quantity
that has magnitude and direction.
Where have we seen these before?
-
Let’s return to the b-ball picture.
a.) What was a position vector in this case?
b.) What was a velocity vector?
c.) How about an acceleration vector?
d.) What do each of these vectors tell us?
Velocity and Acceleration Vectors…
Oh, and Speed too!!!
A Position Vector describes the direction and distance a point/object is from
where it started (think basketball starting point and an arrow meeting its current
position)
A Velocity Vector describes the direction in and speed with which a point/object is
traveling.
An Acceleration Vector describes the direction in and rate at which a
point/object’s velocity is changing.
Before we find position, velocity and acceleration
vectors…let’s get warmed up graphing Vector-Valued
Functions.
Vector-Valued Functions
Vector – Valued Function – A function whose graph is traced out by vectors
(but really acts like a parametric curve!)
G eneral F orm
Let’s trace out a general curve!
f t  , g t 
Sketch r(t) = 𝑡 2 , 𝑡
-Just like parametric
-Remember Orientation
Curves in Space (3D!)
Sketch
4 co s t , 4 sin t , t
Point being….
WE SKETCH VECTOR-VALUED FUNCTIONS THE SAME AS IF
THEY WERE P-METRIC!!
Velocity Vector or Tangent Vector
If
x  t  , y  t  is our position vector at any tim e t, then
x '  t  , y '  t  is the velocity vector at any tim e t.
The velocity or tangent vector represents the instantaneous rate of
change of an object at any specific value of t.
NOTE!
y 't 
x 't 
 slope of tangent to curve
FAMILIAR?
It follows the acceleration vector is:
x "  t  , y "  t  is...you guessed it...
the acceleration vector at any tim e t.
Since you have a velocity vector, by finding its magnitude, you
speed
obtain _____________
at a point.
Speed  v  t  
 x '  t     y '  t  
2
2
Remind you of anything? Let’s check out the conceptual!
A true-to-form FR from an example in our text:
Find the velocity vector, speed, and acceleration vector of a
particle that moves along the following
Let
t 
t 
2 sin   , 2 cos  
2
2
describe the m otion of a p article at any tim e t.
Let’s check out this problem graphically and evaluate each
of the following:
Find the position, velocity and acclerat ion vectors at t   / 2
6  t, t
a.) Sketch the path of a particle moving along the above position
vector.
b.) Sketch the velocity and acceleration vector at (3,3).
r t   e , e
t
t
a.) Sketch the path of a particle moving along the above position
vector.
b.) Sketch the velocity and acceleration vector at (1,1).
A particle moves along a curve described by the equations given
below.
𝑡2, 𝑡3 − 𝑡
Find:
a.) The velocity vector at t = 2
b.) The speed of the particle at t = 2
c.) The acceleration vector at t = 2.
d.) The angle the particle is moving relative to the x-axis at t = 2
A particle moves along a line whose equation is y   3 x  4
1.) Write an equation for the position vector.
2.) Write the velocity and acceleration vectors.
3.) find the speed of the object.
For a particle m oving in the xy-plane its velocity is v  t   4 t  3 t ,1  sin  t ,
3
x  0    2 and y  0   7. Find the particle's position
vector and its position at t  4.
2
A quick trip back to p-metrics with a little vectorage!
A particle m oves along the graph of y  x w ith its x-coordinate
3
changing at a rate of
dx
dt
 2 t . If x 1   2 find:
a.) The particle’s position at t = 3
b.) The speed of the particle at t = 3
c.) Direction of the particle at t = 3
A period shoots Adam Fisher out of a cannon at an initial height of 7 feet
above the PHS parking lot, and he is hopefully “caught” by a daring
volunteer 30 yards directly down the parking lot at a height of 4 feet.
Adam is “released” at an angle of 35 degrees with the horizontal.
a.) Find the speed of Adam when he is released.
b.) Find the maximum height of Adam.
c.) Find the time the “receiver” has to reach the proper reception
position after the cannon releases Adam.
TREAT LIKE A P-METRIC PROBLEM!!!