Question

A graphing calculator is recommended.

An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s(t) = 4 cos(t) + 2 sin(t), t≥ 0, where s is measured in centimeters and t in seconds. (Take the positive direction to be downward.)
(a) Find the velocity and acceleration at time t.

Answer 1

The velocity and acceleration of the mass attached to an elastic band performing SHM are given by v(t) = -4 sin(t) + 2 cos(t) and a(t) = -4 cos(t) - 2 sin(t), respectively, which are the first and second derivatives of the displacement equation s(t) = 4 cos(t) + 2 sin(t).

Step-by-step explanation:

The equation of motion for a mass attached to an elastic band performing simple harmonic motion (SHM) is given by s(t) = 4 cos(t) + 2 sin(t), where s is the displacement in centimeters and t is time in seconds. To find the velocity and acceleration at any given time t, we need to take the first and second derivatives of the displacement function, respectively.

The velocity is the first derivative of displacement with respect to time, which for our function s(t) is:

v(t) = -4 sin(t) + 2 cos(t)

Acceleration is the second derivative of displacement with respect to time, so we differentiate the velocity function:

a(t) = -4 cos(t) - 2 sin(t)

This confirms that the motion can be described by simple harmonic motion, similar to a mass on a spring, and Hooke's law is applicable for such systems.