a mass hanging from a string is set in motion and it\'s ensuing velocity is give
ID: 2856499 • Letter: A
Question
a mass hanging from a string is set in motion and it's ensuing velocity is given by v(t) = -2(pi)(sin)(pi)t for t >=0. Assume that the positive direction is upward and s(0)=2. A). Determine the position function for t>=0 B). At what times does the mass reach its lowest point the first three times? C). At what times does the mass reach its highest point the first three times? a mass hanging from a string is set in motion and it's ensuing velocity is given by v(t) = -2(pi)(sin)(pi)t for t >=0. Assume that the positive direction is upward and s(0)=2. A). Determine the position function for t>=0 B). At what times does the mass reach its lowest point the first three times? C). At what times does the mass reach its highest point the first three times? A). Determine the position function for t>=0 B). At what times does the mass reach its lowest point the first three times? C). At what times does the mass reach its highest point the first three times?Explanation / Answer
A)
Since the position function s(t) is the derivative of the velocity function v(t), you integrate the velocity function and apply the initial condition to determine the constant of integration. Integrating v(t) gives
s(t) = 2pi(cos(pi*t)) / pi + C. = 2(cos(pi*t))
Applying the initial condition gives
s(0) = 2 = 2cos(0) + C ==> C = 0. So the answer is
s(t) = 2(cos(pi*t))
B) & C)
Note that the highest point of the mass occurs when the velocity is zero and where the direction is positive (if the direction is negative, then the mass reaches its lowest point).
By setting the velocity equal to zero, we see that:
-2*sin(t) = 0 ==> t = ±k, where k is an integer.
Since time cannot be negative, t is any positive integer.
Then, since s(t) = 2cos(t) = -2 for every odd integer we see that the mass reaches its lowest point after ever 1, 3, and 5 units of time .
and 2cos(t) = 2 for every even integer, we see that the mass reaches its highest point after ever 2, 4, and 6 units of time (if you count the initial position, it is every 0, 2, and 4 units of time).
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