Download Example 5-7 Terminal Speed

Example 5-7 Terminal Speed
Consider a baseball dropped from rest so that it falls through the air. The downward gravitational force always acts
on the ball, so at the instant it is released it has a downward acceleration of magnitude g. As a result of this acceleration, the ball’s speed increases. According to Equation 5-9, the drag force (which points upward, opposite to the ball’s
motion) also increases. The drag force eventually becomes great enough that it has the same magnitude as the gravitational force. The net force on the ball is then zero, and the ball ceases to accelerate: It has reached its terminal speed, so
it no longer speeds up nor does it slow down. Find the terminal speed of a baseball of mass m = 0.14 kg. Discuss how
the terminal speed would change if the ball were made of iron or if the ball had a lightweight parachute. The coefficient
c equals 1.3 * 10-3 N # s 2 >m2.
Set Up
The free-body diagram shows the two forces
acting on the baseball. We use Equation 5-9 to
find the value of the speed v at which the sum
of these forces is zero, so that the acceleration
is zero and the downward velocity is constant.
s
a Fext on ball
s ball
= Fsdrag on ball + w
sball = 0
= ma
Drag force for larger objects at faster
speeds:
Fdrag = cv2
Fdrag on ball
y
v = terminal
speed
x
(5-9)
wball
Solve
Write Newton’s second law in component form
and solve for the terminal speed vterm.
Newton’s second law in component form applied to the ball:
y: Fdrag on ball + (2wball) = 0
At the terminal speed vterm,
Fdrag on ball = cv 2term so
cv 2term - wball = 0
cv 2term = wball = mg
v 2term =
vterm =
Substitute the numerical values of m and
c for the baseball.
mg
c
mg
A c
Using m = 0.14 kg and c = 1.3 * 10-3 N # s 2 >m2,
vterm =
10.14 kg2 19.80 m>s 2 2
C 1.3 * 10-3 N # s 2 >m2
= 32 m>s
Reflect
What would happen if we simultaneously
dropped a baseball and an iron ball of the
same shape and size? Both objects would
have the same value of the coefficient c, but
the iron ball would have a greater mass m
and so a faster terminal speed. This difference
in terminal speeds is the origin of the notion
that “heavier objects fall faster”—which is a
true statement if we take the drag force into
account. If the baseball and iron ball were
dropped side by side in a vacuum, however,
they would accelerate together and always
have the same speed.
If we added a parachute to the baseball,
the drag force would be much greater for a
given speed, and c would have a much larger
value. The terminal speed of the baseball would
then be greatly reduced. The same thing happens when a skydiver opens her parachute: Her
terminal speed decreases dramatically, from
about 60 m>s before opening the parachute to
about 5 m>s afterward.
vterm =
mg
A c
If two objects have the same
shape and size (and hence the
same value of c) but different
masses, the more massive one
will have a greater value of vterm.
Fdrag on iron ball
Fdrag on baseball
baseball
iron ball
of same
shape
and size
baseball’s wbaseball
terminal
speed
iron ball’s
terminal
speed
wiron ball