sum to infinity

Question

A ball is thrown vertically upwards from the ground. It rises to a height of 10m and then falls and

bounces. After each bounce it rises vertically to 2/3 the height from which it fell.

(i) Find the height to which the ball bounces after the nth impact with the ground.

(ii) Find the total distance travelled by the ball from the first throw to the tenth impact with the

ground.

Answer 1

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Answer 2

a₁ = 20

a₂ = 20(2/3)

a₃ = 20(2/3)(2/3)

...

 

a_n = 20(2/3)^n-1

 

S_n = a₁(1 - rⁿ)/(1 - r)

S₁₀ = 20(1 - (2/3)¹⁰)/  (1 - (2/3))

S₁₀ = 20(1 - 1024/59049)/ (1/3)

S₁₀ = 20(58025/59049)/(1/3)

S₁₀ = 60(58025/59049)

S₁₀= 3481500/59049

S₁₀ ≈ 58.9595

 

After the tenth bounce the ball has traveled approximately 59 meters