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1.5 Additional Applications 31

Suppose that at time zero the object is dropped (not thrown) downward, so v(0) = 0. We now
have an initial value problem for the velocity:
α
2
v = g − v ; v(0) = 0.
m
This differential equation is separable:
1
dv = dt.
g − (α/m)v 2
Integrate to get
m −1 α

tanh v = t + c.
αg mg
This equation involves the inverse of the hyperbolic tangent function tanh(x), which is given by
2x
e − 1
tanh(x) = .
e + 1
2x
Solving for v(t), we obtain

mg αg
v(t) = tanh (t + c) .
α m
Now use the initial condition:

mg αg
v(0) = tanh c = 0.
α m
Since tanh(w) = 0 only for w = 0, this requires that c = 0 and the solution for the velocity is

mg αg
v(t) = tanh t .
α m
This expression yields an interesting and perhaps nonintuitive conclusion. As t →∞,
√
tanh( αg/mt) → 1. This means that

mg
lim v(t) = .
t→∞ α
An object falling under the influence of gravity through a retarding medium will not increase
in velocity indefinitely, even given enough space. It will instead settle eventually into a nearly
√
constant velocity fall, approaching the velocity mg/α as t increases. This limiting value is
called the terminal velocity of the object. Skydivers have experienced this phenomenon.

Sliding Motion on an Inclined Plane A block weighing 96 pounds is released from rest at the
top of an inclined plane of slope length 50 feet and making an angle of π/6 radians with the
√
horizontal. Assume a coefficient of friction of μ = 3/4. Assume also that air resistance acts to
retard the block’s descent down the ramp with a force of magnitude equal to 1/2 of the block’s
velocity. We want to determine the velocity v(t) of the block.
Figure 1.7 shows the forces acting on the block. Gravity acts downward with magnitude
mg sin(u), which is 96sin(π/6) or 48 pounds. Here mg = 96 is the weight (as distinguished
from mass) of the block. The drag due to friction acts in the reverse direction and in pounds is
given by
√
3
−μN =−μmg cos(u) =− (96)cos(π/6) =−36.
4
The drag force due to air resistance is −v/2. The total external force acting on the block has a
magnitude of

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