2. 1/3 points | Previous Answers Zill DiffEQModAp11 3.1.037 My Notes Ask Your Te

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2. 1/3 points | Previous Answers Zill DiffEQModAp11 3.1.037 My Notes Ask Your Teacher Suppose a small cannonball weighing 20 pounds is shot vertically upward, with an initial velocity Vo 290 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 a sdt-v(t) the last differential equation is the same as dydt--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 dv --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)--8000 + 8290e-0.004, (b) Use the result obtained in part (a) to determine the height s(t) of the cannonball measured from ground level 072500 2072500e 0.004ft ft Find the maximum height attained by the cannonbal. (Round your answer to two decimal places.) 1283.14 x ft Submit Answer Save Progress

Explanation / Answer

b) we know, v= ds/dt

hence s(t) can be found out by integrating the function found in part (a) with respect to time,

s= -8000*t - 207250*e-0.04*t + C

at t= 0, s=0,

hence, C= 207250

the expression for s becomes,

s(t) =  -8000*t - 207250*e-0.04*t + 207250

maximum height will be reached at a point of time where velocity is zero,

alternatively maximising the s(t) function, by putting its derivative {v(t)} equal to zero,

we get,

-8000 +8290*e-0.04*t =0

t=0.8902 seconds,

putting this in s(t) to find the maximum height,

s(t) =  -8000*t - 207250*e-0.04*t + 207250

smax = -8000*0.8902 - 207250*e-0.04*0.8902 + 207250

smax = 128.31 ft

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