Question

Group Activity In Exercises 33 and \(34,\) a body is moving in simple harmonic motion with position \(s=f(t)(s\) in meters, \(t\) in seconds). (a) Find the body's velocity, speed, acceleration, and jerk at time \(t\) (b) Find the body's velocity, speed, acceleration, and jerk at time \(t=\pi / 4\) sec. (c) Describe the motion of the body. $$s=\sin t+\cos t$$

Short answer

At any time \(t\), the body's velocity is \(\cos(t) - \sin(t)\), speed is \(|\cos(t) - \sin(t)|\), acceleration is \(-\sin(t) - \cos(t)\), and jerk is \(-\cos(t) + \sin(t)\). At \(t = \pi / 4\) sec, these become 0, 0, \(-\sqrt{2}\), and 0, respectively. At this moment, the body is at rest but not in equilibrium.

Step-by-step solution

Find the body's velocity at time \(t\)

The velocity of the body is the first derivative of the position function. Differentiate \(s = \sin(t) + \cos(t)\) with respect to \(t\). Thus, the velocity \(v = ds/dt = \cos(t) - \sin(t)\).

Find the body's speed at time \(t\)

The speed of the body is the absolute value of the velocity. Hence, speed \(|v| = |\cos(t) - \sin(t)|\).

Find the body's acceleration at time \(t\)

The acceleration of the body is the derivative of the velocity function. Differentiate \(v = \cos(t) - \sin(t)\) with respect to \(t\). Thus, the acceleration \(a = dv/dt = -\sin(t) - \cos(t)\).

Find the body's jerk at time \(t\)

The jerk of the body is the derivative of the acceleration function. Differentiate \(a = -\sin(t) - \cos(t)\) with respect to \(t\). Thus, the jerk \(j = da/dt = -\cos(t) + \sin(t)\)

Find these quantities at \(t = \pi / 4\) sec

Substitute \(t = \pi / 4\) into the equations for velocity, speed, acceleration, and jerk. The velocity \(v = \cos(\pi / 4) - \sin(\pi / 4) = 0\), the speed \(|v| = 0\), the acceleration \(a = -\sin(\pi / 4) - \cos(\pi / 4) = -\sqrt{2}\), and the jerk \(j = -\cos(\pi / 4) + \sin(\pi / 4) = 0\)

Describe the motion of the body

At time \(t = \pi / 4\) sec, the body is momentarily at rest since its velocity is zero, but it is not in equilibrium because there is a net force acting on it, indicated by the non-zero acceleration \(-\sqrt{2}\).