Vertical distance of bouncing ball | Sequences, series and induction | Precalculus | Khan Academy

TL;DR

A ball is dropped from a height of 10 meters, and on each bounce, it reaches half the height of the previous bounce. The total vertical distance traveled by the ball is 30 meters.

Transcript

Let's say that we have a ball that we dropped from a height of 10 meters, and every time it bounces it goes half as high as the previous bounce. So for example, you drop it from 10 meters. The next time its peak height is going to be at 5 meters. So the next time around, on the next bounce, let me draw in that same orange color. And the next bounce... Read More

Key Insights

• 🗺️ The vertical distance traveled by a bouncing ball can be determined by summing the distances traveled during each bounce.
• 💁 The heights of the bounces form a geometric sequence, where each bounce is half the height of the previous bounce.
• 💬 The formula for the sum of an infinite geometric series can be used to find the total distance traveled by the ball.
• 🤒 In this case, the sum of the series is 40, which represents the total traveled distance of 30 meters plus the initial drop of 10 meters.
• 💬 The direction of the ball (up or down) does not affect the total vertical distance traveled.
• 🤪 The distance traveled on each bounce is twice the height reached, as the ball goes up and down.
• 🤒 If the ball were to continue bouncing indefinitely, the distance traveled would approach a total of 40 meters.

Questions & Answers

Q: How high does the ball bounce on the first bounce?

On the first bounce, the ball reaches a height of 5 meters, half the initial height of 10 meters.

Q: What is the pattern in the heights of the bounces?

The heights of the bounces form a geometric sequence, with each bounce being half the height of the previous bounce.

Q: Is the distance traveled on each bounce the same as the height reached?

No, the distance traveled on each bounce is twice the height reached because the ball goes up and down on each bounce.

Q: What is the total vertical distance traveled by the ball?

The total vertical distance traveled by the ball is 30 meters, calculated by summing the distances traveled during each bounce.

Summary & Key Takeaways

• A ball is dropped from a height of 10 meters and each bounce reaches half the height of the previous bounce.

• The total vertical distance traveled by the ball is the sum of the distances traveled during each bounce.

• The total distance is found using the formula for the sum of an infinite geometric series.