Chapter 3.5, Problem 34E

a.

To determine

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

Answer to Problem 34E

  $\begin{array}{l}Attimet\\ Velocity=(\cos t-\sin t),speed=\left|\cos t-\sin t\right|\\ Acceleration=(-\sin t-\cos t)m/{\sec }^2,Jerk=(-\cos t+\sin t)m/{\sec }^3\\ \end{array}$.

Explanation of Solution

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

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

Calculation: Given that $s\text{}\text{}=\text{}\text{}\sin t+\cos t$.

Velocity is rate of change of distance

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

Acceleration is derivative of velocity

  $\begin{array}{l}Acceleration=\frac{dv}{dt}=\frac{d}{dt}(\cos t-\sin t)\\ =(-\cos t-\sin t)m/{\sec }^2\\ speed=\left|Velocity\right|\\ =\left|\cos t-\sin t\right|m/\sec \\ Jerkisderivativeofacceleration\\ Jerk=\frac{dq}{dt}=\frac{d}{dt}(-\sin t-\cos t)=(-\cos t+\sin t)m/{\sec }^3\end{array}$

b.

To determine

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

Answer to Problem 34E

  $\begin{array}{l}Attimet=\frac{\pi }{4}\\ Velocity=0m/\sec ,speed=\left|0\right|=0m/\sec \\ Acceleration=-\sqrt{2}m/{\sec }^2,Jerk=0m/{\sec }^3\end{array}$.

Explanation of Solution

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

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

Calculation: Given that $s\text{}\text{}=\text{}\text{}\sin t+\cos t$.

Velocity is rate of change of distance

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

Acceleration is derivative of velocity

  $\begin{array}{l}Acceleration=\frac{dv}{dt}=\frac{d}{dt}(\cos t-\sin t)\\ =(-\cos t-\sin t)m/{\sec }^2\\ speed=\left|Velocity\right|\\ =\left|\cos t-\sin t\right|m/\sec \\ Jerkisderivativeofacceleration\\ Jerk=\frac{dq}{dt}=\frac{d}{dt}(-\sin t-\cos t)=(-\cos t+\sin t)m/{\sec }^3\end{array}$

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

  $\begin{array}{l}Velocity=\cos \frac{\pi }{4}-\sin \frac{\pi }{4}\\ =\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}=0m/\sec \\ Speed=\left|velocity\right|=\left|0\right|=0m/\sec \\ Acceleration=-\sin \frac{\pi }{4}-\cos \frac{\pi }{4}=-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}=-\sqrt{2}m/{\sec }^2\\ Jerk=-\cos \frac{\pi }{4}+\sin \frac{\pi }{4}=\frac{-1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=0m/{\sec }^3\end{array}$

c.

To determine

To describe: The motion of the body.

Answer to Problem 34E

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

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

Calculation: Given that $s\text{}\text{}=\text{}\text{}\sin t+\cos t$.

Velocity is rate of change of distance

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.