{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. 


a. Determine the position function, for t≥0.

Verified step by step guidance

1. Recall that velocity is the derivative of the position function with respect to time, so we have \(v(t) = s\'(t)\). Given \(v(t) = 2\pi \cos \pi t\), we want to find \(s(t)\) such that \(s\'(t) = 2\pi \cos \pi t\).
2. To find \(s(t)\), integrate the velocity function with respect to \(t\): \(s(t) = \int 2\pi \cos \pi t \, dt\).
3. Use the substitution method for integration: let \(u = \pi t\), so \(du = \pi dt\) or \(dt = \frac{du}{\pi}\). Rewrite the integral in terms of \(u\) and then integrate.
4. After integrating, substitute back to express \(s(t)\) in terms of \(t\). Remember to include the constant of integration \(C\).
5. Use the initial condition \(s(0) = 0\) to solve for the constant \(C\) by plugging in \(t=0\) into your expression for \(s(t)\) and setting it equal to zero.

Key Concepts

Relationship Between Velocity and Position

Velocity is the derivative of the position function with respect to time. To find the position function from velocity, you integrate the velocity function over time, adding a constant of integration determined by initial conditions.

Integration of Trigonometric Functions

Integrating trigonometric functions like cosine involves using standard integral formulas. For example, the integral of cos(ax) is (1/a) sin(ax) + C, where a is a constant. This is essential for finding position from velocity when velocity is given as a cosine function.

Initial Conditions in Differential Calculus

Initial conditions, such as s(0) = 0, are used to determine the constant of integration after integrating. They ensure the solution matches the physical scenario, providing a unique position function that fits the problem's starting point.