A massive object of \(m=5.00 \mathrm{~kg}\) oscillates with simple harmonic motion. Its position as a function of time varies according to the equation \(x(t)=2 \sin ([\pi / 2] t+\pi / 6)\). a) What is the position, velocity, and acceleration of the object at \(t=0 \mathrm{~s}^{2}\) b) What is the kinetic energy of the object as a function of time? c) At which time after \(t=0 \mathrm{~s}\) is the kinetic energy first at a maximum?

Question: Find the position, velocity, and acceleration functions for an object undergoing simple harmonic motion, along with the position, velocity, and acceleration at t=0s. Also, find the kinetic energy function and the first time the kinetic energy reaches its maximum value after t=0s. Answer: The position function is given by: \(x(t) = 2 \sin (\frac{\pi}{2} t + \frac{\pi}{6})\) The velocity function is: \(v(t) = \pi \cos (\frac{\pi}{2} t + \frac{\pi}{6})\) The acceleration function is: \(a(t) = - \frac{\pi^2}{2} \sin (\frac{\pi}{2} t + \frac{\pi}{6})\) At t=0s: Position: \(x(0) = 2 \sin (\frac{\pi}{6}) = 1 \mathrm{~m}\) Velocity: \(v(0) = \pi \cos (\frac{\pi}{6}) = \frac{\pi \sqrt{3}}{2} \mathrm{~m/s}\) Acceleration: \(a(0) = - \frac{\pi^2}{2} \sin (\frac{\pi}{6}) = -\frac{\pi^2}{2} \mathrm{~m/s^2}\) The kinetic energy function is: \(K.E.(t) = \frac{5 \pi^2}{4} \cos^2(\frac{\pi}{2} t + \frac{\pi}{6})\) The first time the kinetic energy reaches its maximum value is at approximately \(t = 0.81\mathrm{~s}\).