A bus moves over a straight level road with a constant acceleration a. A boy in the bus drops a ball out side. The acceleration of the ball w.r.t the bus and the earth are respectively

With respect to Earth, acceleration of ball is g in downward direction as ball is no longer in contact with bus and it is only under the influence of gravity.

$\text{ Acceleration of bus, }{\overrightarrow{\mathrm{a}}}_{\mathrm{b}}=\mathrm{a}\hat{\mathrm{i}}$

$\text{ Acceleration of ball, }{\overrightarrow{\mathrm{a}}}_{\text{ball }}=-\mathrm{g}\hat{\mathrm{j}}$

Acceleration of ball with respect to bus,

$\begin{array}{r}{\overrightarrow{\mathrm{a}}}_{\text{ball, bus }}={\overrightarrow{\mathrm{a}}}_{\text{ball }}-{\overrightarrow{\mathrm{a}}}_{\text{bus }} \\ =-\mathrm{g}\widehat{\mathrm{j}}-\mathrm{a}\widehat{\mathrm{i}} \\ \left|{\overrightarrow{\mathrm{a}}}_{\text{ball,bus }}\right|=\sqrt{(-\mathrm{g}{)}^2+(-\mathrm{a}{)}^2}\\ =\sqrt{{\mathrm{a}}^2+{\mathrm{g}}^2} \end{array}$