Answer
$$\frac{{200}}{3}$$
Work Step by Step
$$\eqalign{
& {\text{Let the trajectory of the ball given by }}y = 0.01x\left( {200 - x} \right) \cr
& {\text{The intersection points with the }}x{\text{ - axis of the function are:}} \cr
& 0.01x\left( {200 - x} \right) = 0 \cr
& x = 0{\text{ or }}x = 200 \cr
& {\text{The average value is given by:}} \cr
& {f_{avg}} = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& {\text{Substituting}} \cr
& {f_{avg}} = \frac{1}{{200 - 0}}\int_0^{200} {0.01x\left( {200 - x} \right)} dx \cr
& {f_{avg}} = \frac{1}{{200}}\int_0^{200} {\left( {2x - 0.01{x^2}} \right)} dx \cr
& {\text{Integrating}} \cr
& {f_{avg}} = \frac{1}{{200}}\left[ {{x^2} - \frac{{{x^3}}}{{300}}} \right]_0^{200} \cr
& {f_{avg}} = \frac{1}{{200}}\left[ {{{\left( {200} \right)}^2} - \frac{{{{\left( {200} \right)}^3}}}{{300}}} \right] - 0 \cr
& {f_{avg}} = \frac{{200}}{3} \cr} $$
