1,: A dropped ball accelerates downwards at a constant rate 9.8 meters per secon

Question

1,: A dropped ball accelerates downwards at a constant rate 9.8 meters per second squared. Set up the differential equation for the height above ground h in meters. Then supposing h(0) = 100 meters, how long does it take for the ball to hit the ground.

2,: A disease is spreading through the country. Let x be the number of people infected. Let the constant S be the total population. Here if one is not infected, then he or she is susceptible to infection. The infection rate dx/dt is proportional to the product of already infected people, x, and the number of susceptible but uninfected people, S x.

(a) Write down the differential equation for x.

(b) Supposing x(0) > 0, that is, some people are infected at time t = 0, what is limt x(t).

(c) Does the solution to part (b) agree with your intuition? Why or why not?

Explanation / Answer

1)

m a + mg = 0

a = h''

or h'' + 9.8 = 0

h'' = -9.8

h' = -9.8 t+ h'(0)

velocity at t= 0 is 0

h'(0)=0

h' = -9.8 t

h = h(0) - 4.9t^2

at t = t* h = 0

or 100 = 4.9*(t*)^2

or

t =4.51754

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