A ball is thrown vertically upwards from the ground. The ball rises to a height of 8 and then falls and bounces. After each bounce it rises to 3/4 of the height of the previous bounce.
a Write down an expression for the height that the ball rises affor the nth impact with the ground.
b Find the total distance that the ball travels from the first throw to the Afth impact with the ground.
a) Since the ball initially rises to a height of 8, we can write the expression for the height the ball reaches after the nth impact with the ground as:
Height(n) = 8 * (3/4)^(n-1)
b) The total distance that the ball travels from the first throw to the Ath impact with the ground can be calculated by summing up the distances of each upward and downward motion.
Let's denote the total distance traveled as D.
The distance traveled during the first upward motion is 8.
For subsequent motions, the distance traveled during each upward motion is 2 times the height reached after the previous bounce.
Therefore, the distance traveled during the upward motion after the nth bounce is 2 * Height(n-1).
The distance traveled during the downward motion after the nth bounce is equal to the height reached after the nth bounce.
So the total distance D can be expressed as:
D = 8 + 2(Height(1) + Height(2) + ... + Height(A))
To find the value of D, we can substitute the expression for Height(n) from part a:
D = 8 + 2((8*(3/4)^0) + (8*(3/4)^1) + ... + (8*(3/4)^(A-1)))
Simplifying this expression will give us the total distance traveled by the ball.
a) To find an expression for the height that the ball rises after the nth impact with the ground, we need to consider the heights of each bounce.
Let's call the first impact with the ground "impact 0".
The height after impact 0 is denoted by H_0 = 8.
For each subsequent impact, the ball rises to 3/4 of the height of the previous bounce.
So, after the first impact (n=1), the height is H_1 = (3/4) * H_0.
After the second impact (n=2), the height is H_2 = (3/4) * H_1.
After the third impact (n=3), the height is H_3 = (3/4) * H_2.
And so on.
In general, we can express the height after the nth impact as:
H_n = (3/4) * H_(n-1)
b) To find the total distance traveled by the ball from the first throw to the nth impact, we need to consider both the upward and downward distances.
Let's denote the distance traveled by the ball during the upward part of each bounce as D_up.
And let's denote the distance traveled by the ball during the downward part of each bounce as D_down.
During the upward part of each bounce, the ball rises to a height H_n, so the distance traveled is equal to H_n.
During the downward part of each bounce, the ball falls from a height H_n to the ground, so the distance traveled is also equal to H_n.
Therefore, the total distance traveled by the ball from the first throw to the nth impact is:
Distance = (H_0 + H_1) + (H_1 + H_2) + ... + (H_(n-1) + H_n)
We can rewrite this in a simplified form:
Distance = H_0 + H_1 + H_1 + H_2 + ... + H_(n-1) + H_n
Simplifying further, we get:
Distance = H_0 + 2 * (H_1 + H_2 + ... + H_(n-1)) + H_n
Substituting the expression for H_1, H_2, ..., H_(n-1), and H_n from part (a), we get:
Distance = 8 + 2 * [(3/4) * H_0 + (3/4) * H_1 + ... + (3/4) * H_(n-1)] + (3/4) * H_n