Transcript AP Calculus BC – 3.6 Chain Rule 1

AP Calculus BC – Chapter 10
Parametric, Vector, and Polar Functions
10.4: Modeling Projectile Motion
Goals: Solve problems involving ideal
projectile motion and projectile motion with
air resistance.
Ideal Projectile Motion:
Assume that a projectile is launched from the
origin at time t=0 into the first quadrant
with an initial velocity v0. If v0 makes an
angle  with the horizontal and it we write v0
for the initial speed |v0|, then
r=(v0cos )ti + ((v0sin )t - ½ gt2)j
is the vector equation for ideal projectile
motion.
Ideal Projectile Motion:
The angle  is the projectile’s launch angle
(firing angle, angle of elevation) and v0 is
the projectile’s initial speed. The projectile’s
range is the distance from the origin to the
point of impact on horizontal ground.
Height, Flight Time and Range:
For ideal projectile motion when an object is
launched from the origin over a horizontal
surface with initial speed v0 and launch angle
:
2

v0 sin  
Maximum height: ymax  2 g
2v0 sin 
t
g
Flight time:
2
Range:
v
R  0 sin 2 
g
Projectile Motion with Linear Drag:
The motion of a projectile with linear drag force launched
from the origin over a horizontal surface at t=0 is
given by





  v0
g
 v0
 kt
 kt
r  1  e cos i   1  e sin    2 1  kt  e kt
k
k
k


j

where the drag coefficient k is a positive constant
representing resistance due to air density, v0 and 
are the projectile’s initial speed and launch angle,
and g is the acceleration of gravity.
Assignment and Notes:

HW 10.4: #1, 5, 6, 13 or 17, 16, 18, 23,
24, 25.
Test Friday, March 16.
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AP Calculus BC exam by next
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