A ball is dropped from a height of 8 feet and bounces. Suppose that each bounce is 7/8 of the height of the bounce before. Thus, after the ball hits the floor for the first time, it rises to a height of 8(7/8) = 7 feet, etc. (Assume g=32 ft/s² and no air resistance.)
A. Find an expression for the height, in feet, to which the ball rises after it hits the floor for the nth time:
B. Find an expression for the total vertical distance the ball has traveled, in feet, when it hits the floor for the first, second, third and fourth times:
first time: D =
second time: D =
third time: D =
fourth time: D =
C. Find an expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the nth time.
A. The height to which the ball rises after it hits the floor for the nth time can be calculated by multiplying the initial height (8 feet) by the ratio of each bounce (7/8). Therefore, the expression for the height is:
Height = 8 * (7/8)^(n-1)
B. The total vertical distance the ball has traveled is the sum of all the heights after each bounce.
First time: D = 8 feet + 7 feet
Second time: D = (8 feet + 7 feet) + (7 feet + 7(7/8) feet)
Third time: D = (8 feet + 7 feet) + (7 feet + 7(7/8) feet) + (7(7/8) feet + 7(7/8)^2 feet)
Fourth time: D = (8 feet + 7 feet) + (7 feet + 7(7/8) feet) + (7(7/8) feet + 7(7/8)^2 feet) + (7(7/8)^2 feet + 7(7/8)^3 feet)
C. To find a closed-form expression for the total vertical distance the ball has traveled when it hits the floor for the nth time, we can use the geometric series formula. The sum of a geometric series is given by:
S = a * (1 - r^n) / (1 - r)
In this case, a = 8 feet (initial height) and r = 7/8 (ratio of each bounce). So the closed-form expression for the total vertical distance is:
D = 8 * (1 - (7/8)^n) / (1 - 7/8)
A. The expression for the height, in feet, to which the ball rises after it hits the floor for the nth time can be found using the following pattern:
Height after 1st bounce = 8 * (7/8)
Height after 2nd bounce = (8 * (7/8)) * (7/8)
Height after 3rd bounce = ((8 * (7/8)) * (7/8)) * (7/8)
...
Height after nth bounce = 8 * (7/8)^(n-1)
B. The total vertical distance traveled by the ball, in feet, when it hits the floor for the first, second, third, and fourth times can be found by summing up the heights reached after each bounce.
First time: D = 8 + 8 * (7/8)
Second time: D = 8 + 8 * (7/8) + (8 * (7/8)) * (7/8)
Third time: D = 8 + 8 * (7/8) + (8 * (7/8)) * (7/8) + ((8 * (7/8)) * (7/8)) * (7/8)
Fourth time: D = 8 + 8 * (7/8) + (8 * (7/8)) * (7/8) + ((8 * (7/8)) * (7/8)) * (7/8) + (((8 * (7/8)) * (7/8)) * (7/8)) * (7/8)
C. The expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the nth time can be found by using the sum of a geometric series formula:
D = 8 * (1 + (7/8) + (7/8)^2 + (7/8)^3 + ... + (7/8)^(n-1))