A mass hanging from a spring is set in motion and its ensuing velocity is given by v(t) = 2𝛑cos𝛑t, for t ≥ 0.

Question

Assume that the positive direction is upward and s(0) = 0.

1. Determine the position function, for t ≥ 0.
2. Graph the position function on the interval [0,4].
3. At what times does the mass reach its low point the first three times?
4. At what time does the mass reach its high point the first three times?

Answer 1

1.

The position function s(t) is the integral of velocity v(t):

s(t) = ∫2πcos(πt)dt = 2sin(πt) + C

The constant C is determined from the requirement that s(0) = 0

0 = 2sin(0) + C

C = 0

3.

The low point is reached when sin(πt) = -1, that is

πt = 2kπ - π/2

t = 2k - 1/2

For some integer k = 1, 2, 3, ...

(k > 0 to ensure t ≥ 0)

So the first three times the minimum is reached at times 1.5, 3.5, 5.5

4.

The high point is reached when sin(πt) = 1, that is

πt = 2kπ + π/2

t = 2k + 1/2

For some integer k = 0, 1, 2, 3, ...

(k ≥ 0 to ensure t ≥ 0)

So the first three times the maximum is reached at times 0.5, 2.5, 4.5