Projectile Motion
with Wind and Air
Resistance
Section 10.4b
Practice Problems
A baseball is hit when it is 3 ft above the ground. It leaves the bat
with initial speed of 152 ft/sec, making an angle of 20 with the
horizontal. At the instant the ball is hit, an instantaneous gust of
wind blows in the horizontal direction directly opposite the
direction the ball is taking toward the outfield, adding a
component of 8.8i (ft/sec) to the ball’s initial velocity.
(a) Find the vector equation (position vector) and the parametric
equations for the path of the baseball.
v 0   v0 cos   i   v0 sin   j  8.8i
 152 cos 20  i  152sin 20  j  8.8i
 152 cos 20  8.8  i  152sin 20  j
Initial position: r0  0i  3j
Practice Problems
A baseball is hit when it is 3 ft above the ground. It leaves the bat
with initial speed of 152 ft/sec, making an angle of 20 with the
horizontal. At the instant the ball is hit, an instantaneous gust of
wind blows in the horizontal direction directly opposite the
direction the ball is taking toward the outfield, adding a
component of 8.8i (ft/sec) to the ball’s initial velocity.
(a) Find the vector equation (position vector) and the parametric
equations for the path of the baseball.
1 2
r   gt j  v 0t  r0
2
2
 16t j  152 cos 20  8.8  ti  152sin 20  tj  3j
 152 cos 20  8.8  ti   3  152sin 20  t  16t
2
j
Practice Problems
A baseball is hit when it is 3 ft above the ground. It leaves the bat
with initial speed of 152 ft/sec, making an angle of 20 with the
horizontal. At the instant the ball is hit, an instantaneous gust of
wind blows in the horizontal direction directly opposite the
direction the ball is taking toward the outfield, adding a
component of 8.8i (ft/sec) to the ball’s initial velocity.
(a) Find the vector equation (position vector) and the parametric
equations for the path of the baseball.
r  152 cos 20  8.8  ti   3  152sin 20  t  16t
Corresponding parametric equations:
x  152 cos 20  8.8  t
y  3  152sin 20  t  16t
2
2
j
Practice Problems
(b) How high does the baseball go, and when does it reach
maximum height?
y  3  152sin 20  t  16t
dy
 152sin 20   32t  0
dt
2
152sin 20
t
32
 1.625sec
 152sin 20 
 152sin 20 
ymax  3  152sin 20  
  16 

32
32




The baseball reaches a maximum  45.229ft
height of about 45.229 ft after
approximately 1.625 sec.
2
Practice Problems
(c) Assuming the ball is not caught, find its range and flight time.
y  3  152sin 20  t  16t  0 t  3.306sec
2
x  R  152 cos 20  8.8  3.306   443.102ft
The baseball is in flight for approximately
3.306 sec, and travels a horizontal distance
of about 443.102 ft during that time.
Confirm these answers graphically!!!
Exploration 1
Suppose the baseball from Example 4 is hit toward the rightcenterfield fence, 400 ft from the batter.
1. If the fence is 15 ft high, is the hit a home run (does it clear the
fence in flight)?
Time for the ball to reach the fence:
152 cos 20  8.8 t  400
t  2.984sec
Height of the ball at this time:
y  3  152sin 20  2.984   16  2.984 
 15.647ft
2
The baseball clears the fence
by less than a foot!
Exploration 1
Suppose the baseball from Example 4 is hit toward the rightcenterfield fence, 400 ft from the batter.
2. Using graphical methods, find the range and the flight time if
the angle at which the ball comes off the bat is:
 a  25,  b  30,  c  45
Graph in [0, 500] by [–20, 60]
Angle (degrees)
25
30
45
Range (ft)
 523.707  588.279  665.629
Flight time (sec)
 4.061
 4.789
 6.745
Projectile Motion with Linear Drag
The main force affecting the motion of a projectile, other than
gravity, is air resistance. This slowing force is drag force and it
acts in a direction opposite to the velocity of the projectile:
Velocity
Drag force
Gravity
Note: For projectiles moving through the air at relatively low
speeds, the drag force is (very nearly) proportional to the speed
(to the first power), and so is called linear.
Projectile Motion with Linear Drag
Equations for the motion of a projectile with linear drag force
launched from the origin over a horizontal surface at t = 0:
v0
 kt
Vector form: r 
1  e   cos   i 

k
g
 v0
 kt
 kt 
1

e
sin


1

kt

e
j






2
k

k


v0
 kt
Parametric form: x 
1  e  cos 

k
v0
g
 kt
 kt
y  1  e   sin    2 1  kt  e 
k
k
where the drag coefficient k is a positive constant representing
resistance due to air density.
Exploration 2
Suppose the baseball from Example 4 is hit with linear drag to
slow it down instead of wind. For the temperature and humidity
on game day, the drag coefficient k is 0.05.
1. Find and graph the parametric equations for the path of the
baseball.
152
0.05t
x
1 e
cos 20


0.05
152
0.05t
y  3
1 e
sin 20


0.05
32
0.05t

1  0.05t  e

2 
0.05
Graph in [0, 450] by [–20, 60]
Exploration 2
Suppose the baseball from Example 4 is hit with linear drag to
slow it down instead of wind. For the temperature and humidity
on game day, the drag coefficient k is 0.05.
2. How high does the baseball go, and when does it reach this
maximum height?
Estimate answers graphically…
The ball reaches a maximum height of
about 43.07 ft when t is about 1.56 sec.
3. Assuming that the ball is not caught, find the range and flight
time of the baseball.
Estimate answers graphically…
The range is about 425.47 ft and the
flight time is about 3.23 sec.