Suppose a small cannonball weighing 98 n is shot vertically upward, with an initial velocity
v0 = 88 m/s.
the answer to the question "how high does the cannonball go? " depends on whether we take air resistance into account. if air resistance is ignored and the positive direction is upward, then a model for the state of the cannonball is given by
d2s/dt2 = −g
(equation (12) of section 1.3). since
ds/dt = v(t)
the last differential equation is the same as
dv/dt = −g,
where we take
g = 9.8 m/s2.
if air resistance is incorporated into the model, it stands to reason that the maximum height attained by the cannonball must be less than if air resistance is ignored.
(a) assume air resistance is proportional to instantaneous velocity. if the positive direction is upward, a model for the state of the cannonball is given by m
dv
dt
= −mg − kv, where m is the mass of the cannonball and
k > 0
is a constant of proportionality. suppose
k = 0.0025
and find the velocity
v(t)
of the cannonball at time t.
find the maximum height attained by the cannonball. (round your answer to two decimal places.)