Answer:
a) 22π cm/s; on the odd half-second (t=0.5, 1.5, 2.5, 3.5, 4.5); at the rest position; zero
b) 22 cm from the rest position; zero
Step-by-step explanation:
Undamped simple harmonic motion is not complicated. Acceleration is a maximum where the applied force is a maximum, at the extremes of position. Since the position is extreme, the velocity is zero at those points. All of the energy is potential energy.
The speed is a maximum when the object is at the rest position, There is no applied force at that point, and no acceleration. All of the energy has been transformed to kinetic energy.
<u>Part A</u>:
The cart's velocity is given by the derivative of the position:
s'(t) = -22π·sin(πt)
This has a maximum magnitude (speed) of 22π ≈ 69.1 cm/s, as you have noted.
The speed is a maximum at the rest position. The cart is there on each odd quarter-period, at t=0.5, 1.5, 2.5, 3.5, 4.5 seconds.
The cart's acceleration is given by the derivative of the velocity:
s'' = -22π²·cos(πt)
On the odd quarter-period, the acceleration is zero.
<u>Part B</u>:
Acceleration is greatest when position is greatest (both are cosine functions). The speed of the cart is zero then (it is a sine function). The sine is at an extreme when the cosine is zero, and vice versa.
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The attached graph shows position, velocity, and acceleration (color coded).