A ball is dropped on the ground. The time interval is between the first and second bounce is 1.2s. The time interval between the second and third bounce is 0.9s. Assume that successive time intervals form a geometric progression with infinite number of terns. Hence find the total time for which the ball bounce
Let's denote the time interval between the first and second bounce as "t" and the common ratio between the time intervals as "r".
According to the problem, we know:
t * r = 1.2 (Equation 1)
t * r^2 = 0.9 (Equation 2)
Dividing Equation 2 by Equation 1, we get:
r = 0.9/1.2 = 0.75
Now we can use Equation 1 to solve for "t":
t = 1.2/r = 1.2/0.75 = 1.6
So the time interval between each bounce is:
t, tr, tr^2, tr^3, ...
Plugging in the values of "t" and "r", we can see that:
t = 1.6
tr = 1.2
tr^2 = 0.9
tr^3 = 0.675
tr^4 = 0.50625
...
The sum of an infinite geometric series is given by:
S = a/(1 - r)
where "a" is the first term in the series. In this case, "a" is "t" since that is the time interval between the first and second bounce.
Plugging in the values, we get:
S = 1.6/(1 - 0.75) = 6.4 seconds
Therefore, the total time for which the ball bounces is 6.4 seconds.
To find the total time for which the ball bounces, we need to find the sum of the infinite geometric series formed by the time intervals.
Let's denote the first time interval as 'a' and the common ratio as 'r'.
From the given information, we have:
Second bounce - First bounce = 1.2s.
Third bounce - Second bounce = 0.9s.
Thus, we have the following equations:
a * r = 1.2 (equation 1)
a * r^2 = 0.9 (equation 2)
Now, let's solve these equations to find the values of 'a' and 'r'.
Divide equation 2 by equation 1:
(a * r^2) / (a * r) = 0.9 / 1.2
r = 0.9 / 1.2
r = 0.75
Substituting the value of 'r' in equation 1:
a * 0.75 = 1.2
a = 1.2 / 0.75
a = 1.6
Now that we have the values of 'a' and 'r', we can find the sum of the infinite geometric series using the formula:
Sum = a / (1 - r)
Substituting the values:
Sum = 1.6 / (1 - 0.75)
Sum = 1.6 / 0.25
Sum = 6.4
Therefore, the total time for which the ball bounces is 6.4 seconds.