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
Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
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- Find the sum of the G.P.:
$5/11, 5/121, 5/1331, 5/14641, ...$ to $n$ terms. The value of $1 + (\frac{1}{2^1} + \frac{1}{3}) + (\frac{1}{2^2} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}) + ... + (\frac{1}{2^9} + ... + \frac{1}{1023})$ lies between
- An ant starts at the origin and moves along the $y$-axis and covers a distance $l$. This is its first stage in its journey. Every subsequent stage requires the ant to turn right and move a distance which is half of its previous stage. What would be its coordinates at the end of its $5^{\text{th}}$ stage?
