Suppose you drop a rock off a cliff of height h As it falls

Suppose you drop a rock off a cliff of height h . As it falls, you snap a million photographs, at random intervals. On each picture you measure the distance that the rock has fallen.

(a) Calculate and plot the probability density as a function of x?

(b) What is the average of all these distances?

Solution

a. According to classical mechanics, we have x(t)=1/2*g*t^2, T=sqrt(2h/g)

b. The probability between and is, dt/T=dx/gt*sqrt(g/2h)=1/(2*sqrt(hx))*dx. So P(x)=1/(2*sqrt(hx)),(0<=x<=h)

Integrating p(x) we get 1/(2*sqrt(h))*2*x^1/2 for 0<x<h which result to 1(normalization condition)

then, the average distance is, (x)=intergration of x*P(x)dx,for 0<x<h=1/(2*sqrt(h))*2/3*x^3/2 for 0<x<h which result to h/3 and that\'s the answer