(I) A ball is dropped from a height of 10 m and bounces continuously. With each successive bounce, the ball reaches a height that is \( 50 \% \) of the previous height. If this motion continues indefinitely, what is the total vertical distance travelled by the ball over its entire journey?

When the ball is dropped from a height of 10 m, it travels down 10 m initially. After that, it bounces up to 5 m (which is \( 50 \% \) of 10 m) and then falls the same distance of 5 m down again. This process continues indefinitely, forming a geometric series for the heights reached by the bounces. The total distance can be calculated as follows: 1. The first drop is 10 m. 2. The ball bounces up to 5 m and then drops down 5 m. 3. The second bounce reaches 2.5 m and then drops down again, continuing this pattern. The upward distances after the first drop can be structured as: - Up: 5 m + 2.5 m + 1.25 m + ... (a geometric series with \( a = 5 \) m and \( r = 0.5 \)) - Down: 5 m + 2.5 m + 1.25 m + ... (the same geometric series as the upward distances) Now, calculating the geometric series sum: \( S_{\text{upward}} = \frac{a}{1 - r} \) The height of the first bounce series: \[ S = \frac{5}{1 - 0.5} = \frac{5}{0.5} = 10 \text{ m}. \] Now calculating the total distance: \[ \text{Total Distance} = \text{Initial Drop} + \text{Total Upward Distance} + \text{Total Downward Distance} \] \[ = 10 + 10 + 10 = 30 \text{ m}. \] Thus, the total vertical distance traveled by the ball over its entire journey is \( 30 \text{ m} \).