{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0.

c. At what times does the mass reach its low point the first three times?

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Understand that the low points of the oscillating mass correspond to the minimum positions in its motion. Since velocity \(v(t)\) is the derivative of position \(s(t)\), the low points occur when the velocity changes from negative to positive, which means the velocity is zero and the acceleration is positive.

Set the velocity function equal to zero to find critical points: \(v(t) = 2\pi \cos(\pi t) = 0\). Solve for \(t\) such that \(\cos(\pi t) = 0\).

Recall that \(\cos(\theta) = 0\) at \(\theta = \frac{\pi}{2} + n\pi\) for integers \(n\). Substitute \(\theta = \pi t\) to get \(\pi t = \frac{\pi}{2} + n\pi\), which simplifies to \(t = \frac{1}{2} + n\) where \(n\) is an integer \(\geq 0\) because \(t \geq 0\).

Determine which of these times correspond to the mass reaching its low point by checking the sign of the acceleration \(a(t) = v'(t)\). Compute \(a(t) = \frac{d}{dt} v(t) = \frac{d}{dt} (2\pi \cos(\pi t))\) and evaluate \(a(t)\) at each critical time to confirm it is positive (indicating a minimum).

List the first three times \(t\) (starting from \(t \geq 0\)) where \(v(t) = 0\) and \(a(t) > 0\). These times are when the mass reaches its low point for the first three occurrences.

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Relationship Between Velocity and Position in Oscillatory Motion

In oscillatory motion, velocity is the derivative of position with respect to time. To find when the mass reaches its low point, we analyze the velocity and position functions. The low point corresponds to a local minimum in position, which occurs when velocity changes sign from negative to positive.

Critical Points and Extrema of a Function

Critical points occur where the derivative (velocity) is zero or undefined. For oscillating systems, these points indicate potential maxima or minima in position. Determining whether a critical point is a minimum involves checking the sign changes of velocity or using the second derivative test.

Solving Trigonometric Equations

The velocity function involves cosine, so finding times when velocity is zero requires solving trigonometric equations like cos(πt) = 0. Understanding the periodicity and zeros of cosine helps identify specific time values corresponding to turning points in motion.

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. c. At what times does the mass reach its low point the first three times?