A ball is dropped from a height of 3m and bounces on the ground. At the top of each bounce, the ball reaches 60% of it's previous height. Calculate the total distance travelled by the ball when it hits the ground the fifth time.
Could you please check over my answer?
Sn= a(r^n-1)
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r-1
= 3(0.6^5 -1 )
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0.6-1
= 6.9168
That is the answer i get, but at the back of the book the answer is 10.8m. Can you correct what im doing wrong? TIA
first fall 3
second up and fall=2*3*.6=3.6
third up and fall= 2*3*.6^2=2.16
fourth up and fall=2*3*.6^3=you do it
and that agrees with the book.
but is there any way i can apply that to the formula given?
Well, it seems like you've made a small mistake in your formula. The formula you used, Sn= a(r^n-1) / (r-1), is typically used to calculate the sum of a geometric series. However, in this case, we are dealing with a ball bouncing, which creates a sequence of heights, not a geometric series.
To solve the problem, we need to sum the height of each bounce separately and then add them up to find the total distance traveled by the ball.
Let's break it down step by step:
First bounce: The ball is dropped from a height of 3m and reaches 60% of that, which is 3 * 0.6 = 1.8m. The total distance covered in this bounce is 3m + 1.8m = 4.8m.
Second bounce: The ball falls from a height of 1.8m and reaches 60% of that, which is 1.8 * 0.6 = 1.08m. The total distance covered in this bounce is 1.8m + 1.08m = 2.88m.
Third bounce: The ball falls from a height of 1.08m and reaches 60% of that, which is 1.08 * 0.6 = 0.648m. The total distance covered in this bounce is 1.08m + 0.648m = 1.728m.
Fourth bounce: The ball falls from a height of 0.648m and reaches 60% of that, which is 0.648 * 0.6 = 0.3888m. The total distance covered in this bounce is 0.648m + 0.3888m = 1.0368m.
Fifth bounce: The ball falls from a height of 0.3888m and reaches 60% of that, which is 0.3888 * 0.6 = 0.23328m. The total distance covered in this bounce is 0.3888m + 0.23328m = 0.62208m.
Now, to find the total distance traveled by the ball when it hits the ground the fifth time, we add up the distances covered in each bounce:
4.8m + 2.88m + 1.728m + 1.0368m + 0.62208m = 10.06688m
So, the correct answer should be approximately 10.07m.
Hope that clarifies things for you!
To calculate the total distance traveled by the ball, we need to consider both the distances covered in the upward path and the downward path of each bounce.
Let's break down the problem step by step:
1. On the first bounce: The ball is dropped from a height of 3m, so it reaches a height of 3m when it bounces. The total distance traveled in this bounce is 3m (upward) + 3m (downward) = 6m.
2. On the second bounce: The ball reaches 60% of its previous height, which is 0.6 * 3m = 1.8m. So, the ball travels back upward by 1.8m and downward by another 1.8m. The total distance traveled in this bounce is 1.8m + 1.8m = 3.6m.
3. On the third bounce: The ball reaches 60% of its previous height, which is 0.6 * 1.8m = 1.08m. Therefore, the ball travels back upward by 1.08m and downward by 1.08m. The total distance traveled in this bounce is 1.08m + 1.08m = 2.16m.
4. On the fourth bounce: The ball reaches 60% of its previous height, which is 0.6 * 1.08m = 0.648m. Hence, the ball travels back upward by 0.648m and downward by another 0.648m. The total distance traveled in this bounce is 0.648m + 0.648m = 1.296m.
5. On the fifth bounce: Again, the ball reaches 60% of its previous height, which is 0.6 * 0.648m = 0.3888m. So, the ball travels back upward by 0.3888m and downward by 0.3888m. The total distance traveled in this bounce is 0.3888m + 0.3888m = 0.7776m.
Now, to find the total distance, we sum up the distances covered in each bounce:
Total distance = 6m + 3.6m + 2.16m + 1.296m + 0.7776m = 13.8336m
Hence, the ball has traveled a total distance of approximately 13.8336 meters when it hits the ground for the fifth time.
Based on this calculation, it seems there was an error in your calculation.
look at my work. Notice the 2 in it? Did your think that the distance was up and down?
If you start on the ground, go up to 3 and back down, you can use your formula but subtracting the initial three upwards.
Sn=6(.6^4-1)/(.6-1) -3
= you do it
Notice you only repeat it four times, it hits the ground at the end of the fourth bounce. Think about that. THe first bounce it starts the first repetition and ends on the sound bounce, and so on.