(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?

To find the total vertical distance traveled by the ball over its entire journey, we need to calculate the sum of the distances traveled during each bounce. Given: - Initial height = 10 m - Each bounce reaches a height that is 50% of the previous height Let's denote the height after the first bounce as \( h_1 \), the height after the second bounce as \( h_2 \), and so on. The height after the first bounce is 50% of the initial height: \[ h_1 = 0.5 \times 10 = 5 \, \text{m} \] The height after the second bounce is 50% of the height after the first bounce: \[ h_2 = 0.5 \times h_1 = 0.5 \times 5 = 2.5 \, \text{m} \] We can see that the height after each bounce is decreasing by a factor of 0.5. The total vertical distance traveled by the ball over its entire journey is the sum of the distances traveled during each bounce. This forms a geometric series with the first term \( a = 10 \) m and the common ratio \( r = 0.5 \). The sum of an infinite geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] Substitute the values of \( a = 10 \) m and \( r = 0.5 \) into the formula to find the total vertical distance traveled by the ball over its entire journey. Calculate the value by following steps: - step0: Calculate: \(\frac{10}{\left(1-0.5\right)}\) - step1: Remove the parentheses: \(\frac{10}{1-0.5}\) - step2: Subtract the numbers: \(\frac{10}{0.5}\) - step3: Convert the expressions: \(\frac{10}{\frac{1}{2}}\) - step4: Multiply by the reciprocal: \(10\times 2\) - step5: Multiply the numbers: \(20\)