Each bounce is $2/3$ times the height of the previous bounce. The first bounce reaches $2$ meters, the second $4/3$, the third $8/9$, the fourth $16/27$, and the fifth $32/81$. Half of $81$ is $40.5$, so the ball does not reach the required height on bounce $\boxed{\textbf{(C)}\ 5}$.
A ball is dropped from a height of $3$ meters. On its first bounce it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters?
$\textbf{(A)}\  3 \qquad \textbf{(B)}\  4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$