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Part A - The average value of v(t) over the interval (0, π/2) is 6/π
Part B - The displacement of the yo-yo from time t = 0 to time t = π is 0 m
Part C - The total distance the yo-yo travels from time t = 0 to time t = π is 6 m.
<h3>Part A: Find the average value of v(t) on the interval (0, π/2)</h3>The average value of a function f(t) over the interval (a,b) is
So, since velocity at time t is given by v(t) = 3cos(t) for time t ≥ 0. Its average value over the interval (0, π/2) is given by
Since v(t) = 3cost, we have
So, the average value of v(t) over the interval (0, π/2) is 6/π
<h3>Part B: What is the displacement of the yo-yo from time t = 0 to time t = π?</h3>To find the displacement of the yo-yo, we need to find its position.
So, its position x = ∫v(t)dt
= ∫3cos(t)dt
= 3∫cos(t)dt
= 3sint + C
Given that at t = 0, x = 3. so
x = 3sint + C
3 = 3sin0 + C
3 = 0 + C
C = 3
So, x(t) = 3sint + 3
So, its displacement from time t = 0 to time t = π is
Δx = x(π) - x(0)
= 3sinπ + 3 - (3sin0 + 3)
= 3 × 0 + 3 - 0 - 3
= 0 + 3 - 3
= 0 + 0
= 0 m
So, the displacement of the yo-yo from time t = 0 to time t = π is 0 m
<h3>Part C: Find the total distance the yo-yo travels from time t = 0 to time t = π. (10 points)</h3>The total distance the yo-yo travels from time t = 0 to time t = π is given by
So, the total distance the yo-yo travels from time t = 0 to time t = π is 6 m.
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