uppose a small cannonball weighing 16 pounds is shot vertically upward, with an initial velocity v₀ = 250 ft/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 = 32 ft/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 mdv

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.

v(t) =

(b) Use the result obtained in part (a) to determine the height s(t)

of the cannonball measured from ground level.

s(t) =

ft

Find the maximum height attained by the cannonball. (Round your answer to two decimal places.)

ft