Question

A ball is thrown straight up and falls straight back down. Find the probability density function$f\left(h\right)$ so that $f\left(h\right)dh$is the probability of finding it between height $h$and $h+dh$. Hint: Look at Example $3$.

Short answer

The probability density function is $f\left(h\right)=\frac{1}{2\sqrt{l}\sqrt{l-h}}$.

Step-by-step solution

Given Information

A ball is thrown straight up and falls straight back down.

Definition of the probability density function.

a continuous random variable,whose integral across an interval offers the likelihood that the variable's value falls inside the same interval.

Find the probability density function.

The velocity is given as$v^2=2g(l-h)$.

The time is given below.

$\begin{array}{c}f\left(h\right)h\propto \frac{1}{\sqrt{l-h}}\\ f\left(h\right)=\frac{c}{\sqrt{l-h}}\end{array}$

Find the value of c is given below.

$\begin{array}{c}{\int }_{-\infty }^{\infty }f\left(x\right)dx=1\\ {\int }_0^l\frac{c}{\sqrt{l-h}}=2c\sqrt{l}\\ c=\frac{1}{2\sqrt{l}}\end{array}$

The probability density function is $f\left(h\right)=\frac{1}{2\sqrt{l}\sqrt{l-h}}$