Asmall frictionless cart is attached to a wall by a spring. it is pulled 22 cm from its rest position, released at time t = 0, and allowed to roll back and forth for 5 seconds. its position at time t is s = 22 cos (pi*t).a. what is the cart's maximum speed? when is the cart moving that fast? where is it then? what is the magnitude of the acceleration then? b. where is the cart when the magnitude of the acceleration is greatest? what is the cart's speed then? i cannot figure out part b. i have part a: which is 69.1 cm/sec

  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.

Part A:

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.

Part B:

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).

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step-by-step explanation: