Ball Bounce Distance Calculation

This problem requires calculating the total vertical distance a ball travels between the 1st and 5th bounces. The ball starts from 100 m and rebounds to half its previous height after each impact.

Bounce Heights Calculation

The initial drop height is $h_0 = 100$ m. The height the ball reaches after each successive bounce decreases.

• Height reached after 1st bounce: $h_1 = h_0 \times \frac{1}{2} = 100 \times \frac{1}{2} = 50$ m
• Height reached after 2nd bounce: $h_2 = h_1 \times \frac{1}{2} = 50 \times \frac{1}{2} = 25$ m
• Height reached after 3rd bounce: $h_3 = h_2 \times \frac{1}{2} = 25 \times \frac{1}{2} = 12.5$ m
• Height reached after 4th bounce: $h_4 = h_3 \times \frac{1}{2} = 12.5 \times \frac{1}{2} = 6.25$ m

Distance Travelled Between Bounces

The vertical distance travelled between two consecutive bounces ($n$ and $n+1$) consists of the ball rising to height $h_n$ and falling back down $h_n$. The total distance for this segment is $2 \times h_n$. We need the sum of distances from the 1st bounce up to the 5th bounce impact.

• Distance covered after 1st bounce until 2nd bounce impact: $2 \times h_1 = 2 \times 50 = 100$ m
• Distance covered after 2nd bounce until 3rd bounce impact: $2 \times h_2 = 2 \times 25 = 50$ m
• Distance covered after 3rd bounce until 4th bounce impact: $2 \times h_3 = 2 \times 12.5 = 25$ m
• Distance covered after 4th bounce until 5th bounce impact: $2 \times h_4 = 2 \times 6.25 = 12.5$ m

Total Vertical Distance Summation

The total vertical distance travelled between the first and fifth bounces is the sum of the distances calculated for each segment:

Total Distance = $(2 \times h_1) + (2 \times h_2) + (2 \times h_3) + (2 \times h_4)$

Total Distance = $100 + 50 + 25 + 12.5$ m

Total Distance = $187.5$ m

To express this as a fraction:

Total Distance = $\frac{1875}{10} = \frac{375}{2}$ m