In this question we investigate a falling object of mass M (which we think of as

Question

In this question we investigate a falling object of mass M (which we think of as a point mass). Let x(t) denote the (vertical) position of the object. The only force acting on the object is gravity, which means we assume x satisfies the ODE, M d^2x/dt^2 = -gM, where g is a constant (approximately equal to 10m/s^2). (a) What is the general solution to this ODE? (b) You drop the object from a 100m cliff. How long does it take to hit the ground? Does this depend on M (c) You are on the 100m cliff and throw the ball up into the air with a velocity of 1m/s. How long does it take the object to hit the ground? (d) What ODE does v = dx/dt solve? (e) What is the limit as t approaches infinity of v (t)?

Explanation / Answer

a)

M cancels out and we get

x''=-g

x'=-gx+A

x=-gt^2/2+At+B

b)

x(0)=100

dx/dt(0)=1

x(0)=B=100

dx/dt=-gt+A

Hence, A=1

x=-gt^2/2+t+100

We need, x=0=-gt^2/2+t+100

-5t^2+t+100=0

5t^2-t-100=0

Solving gives

t=(1+sqrt{1+4*100*5})/(2*5)=(1+sqrt{2000})/10

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