a ball is dropped from a height of 40cm onto a horizontal surface and bounces in a vertical line until it eventually stops on the surface.
the ball bounces in such a way that it loses ten percent of its height for each bounce and will continue to do this until it comes to rest on the surface.
what is the total distance travelled by the ball before coming to rest on the surface?how do i calculate this?
The sequence of successive heights can be written as
h---40 + 40(.9) + 40(.9^2) + 40(.9^3) +........ or
40(1 + .9 + .9^2 + .9^3 + ......
The limit of the converging series Sn = 1 + r + r^2 + r^3 + r^4 + = 1/(1 - r) -1 < r < 1
Thus, the total distance traveled becomes D = 40(1)/(1 - .9) = 40/.1 = 400 cm.
Why did the ball become a mathematician? Because it knew how to calculate its bouncing distance! The total distance traveled by the ball before coming to rest on the surface is 400 cm. This can be calculated using the formula D = 40/(1 - 0.9), which simplifies to D = 400 cm. So, make sure you have your measuring tape ready to witness this impressive bouncing journey!
To calculate the total distance traveled by the ball before coming to rest on the surface, you can use the formula for the sum of an infinite geometric series. The sequence of heights can be expressed as 40 + 40(.9) + 40(.9^2) + 40(.9^3) + ..., where 40 is the initial height and each subsequent term is 90% of the previous term.
The sum of an infinite geometric series is given by the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. In this case, a = 40 and r = 0.9.
Plugging the values into the formula:
S = 40 / (1 - 0.9) = 40 / 0.1 = 400 cm.
Therefore, the total distance traveled by the ball before coming to rest on the surface is 400 cm.
To calculate the total distance traveled by the ball before coming to rest on the surface, we can use the formula for the sum of an infinite geometric series.
In this case, the first term (a) is 40 cm (the initial height), and the common ratio (r) is 0.9 (since the ball loses 10% of its height with each bounce).
The sum of the infinite geometric series can be calculated using the formula: S = a/(1 - r)
Plugging in the values, we get: S = 40/(1 - 0.9)
Simplifying the expression, we have: S = 40/0.1 = 400
Therefore, the total distance traveled by the ball before coming to rest on the surface is 400 cm.