Chapter 3.5, Problem 33E

a.

To determine

To find: The body’s velocity, speed, acceleration, and jerk at time $t$.

Answer to Problem 33E

  $\begin{array}{l}Attimet\\ Velocity=-2\cos tm/\sec ,speed=2\cos tm/\sec \\ Acceleration=2\sin tm/{\sec }^2,Jerk=2\cos tm/\sec \end{array}$

Explanation of Solution

Given information: Given a body is moving in simple harmonic motion with position

  $\begin{array}{l}s=f(t)\\ s=2-2\sin t\end{array}$.

Calculation: Given that $s=2-2\sin t$.

  $\begin{array}{l}Velocity=\frac{ds}{dt}=\frac{d}{dt}(2-2\sin t)\\ =0-2\cos t\\ =-2\cos tm/\sec \end{array}$

Derivative of distance is velocity

  $\begin{array}{l}speed=\left|Velocity\right|\\ =\left|-2\cos t\right|=2\cos tm/\sec \end{array}$

Acceleration is the rate of change of velocity

  $acceleration=\frac{dv}{dt}=\frac{d}{dt}(-2\cos t)=2\sin tm/{\sec }^2$

Jerk is the rate of change of acceleration

  $Jerk=\frac{dq}{dt}=\frac{d}{dt}(2\sin t)=2\cos tm/{\sec }^3$

b.

To determine

To find: the body’s velocity, speed, acceleration, and jerk at time $t=\frac{\pi }{4}sec$.

Answer to Problem 33E

  $\begin{array}{l}Attimet=\frac{\pi }{4}\\ Velocity=-\sqrt{2}m/\sec ,speed=\sqrt{2}m/\sec \\ Acceleration=\sqrt{2}m/{\sec }^2,\end{array}$

Explanation of Solution

Given information: we are given a body is moving in simple harmonic motion with position

  $\begin{array}{l}s=f(t)\\ s=2-2\sin t\end{array}$

Calculation: We are given that $s=2-2\sin t$

(a) $\begin{array}{l}Velocity=\frac{ds}{dt}=\frac{d}{dt}(2-2\sin t)\\ =0-2\cos t\\ =-2\cos tm/\sec \end{array}$

Derivative of distance is velocity

  $\begin{array}{l}speed=\left|Velocity\right|\\ =\left|-2\cos t\right|=2\cos tm/\sec \end{array}$

Acceleration is the rate of change of velocity

  $acceleration=\frac{dv}{dt}=\frac{d}{dt}(-2\cos t)=2\sin tm/{\sec }^2$

Jerk is the rate of change of acceleration

  $Jerk=\frac{dq}{dt}=\frac{d}{dt}(2\sin t)=2\cos tm/{\sec }^3$

Now find velocity, speed, acceleration, and jerk at $t=\frac{\pi }{4}$

  $\begin{array}{l}Velocity=-2\cos t\\ =-2\cos \frac{\pi }{4}\\ =-2\times \frac{1}{\sqrt{2}}=-\sqrt{2}m/\sec \\ Speed=\left|-\sqrt{2}\right|=\sqrt{2}m/\sec \\ Acceleration=2\sin \frac{\pi }{4}=2\times \frac{1}{\sqrt{2}}=\sqrt{2}m/{\sec }^2\\ Jerk=2\cos \frac{\pi }{4}=2\times \frac{1}{\sqrt{2}}=\sqrt{2}m/{\sec }^3\end{array}$

c.

To determine

To describe: The motion of the body.

Answer to Problem 33E

Motion is the phenomenon in which an object changes its position over time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and time.

Explanation of Solution

Given information: we are given a body is moving in simple harmonic motion with position

  $\begin{array}{l}s=f(t)\\ s=2-2\sin t\end{array}$

Calculation: Given that $s=2-2\sin t$.

Motion is the phenomenon in which an object changes its position over time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and time.