Q68P - Mathematical Methods in Physical Sciences

Question

Question: Find v and a if $\mathit{z}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{t}\mathbf{+}\mathit{i}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{t}$, can you describe the motion.

Step-by-Step Solution

Answer

The value of a and v is given below:

 v = 2, a = 4

The motion is circular.

Step 1: Given Information

The function $z=\cos 2t+i \sin 2t$.

Step 2: Definition of the Complex number

A complex number can be dictated as:

 z = x + iy

 Where z is the complex number, x and y are real numbers, and i is known as iota, whose value is $\sqrt{-1}$  .

The modulus of a complex number can be calculated as:

 $\left|z\right|=\sqrt{x^2+y^2}$

Step 3: Find the value of x and y

The function $z=cos2t+i sin2t$ .

      

The value of x and y is given below:

 $$\begin{gathered} x=\cos 2t \\ y=\sin 2t \end{gathered}$$

Step 4: Find the value of the velocity

The formula for velocity is given below:

 $$\begin{gathered} v\left(t\right)=\left|\frac{dz}{dt}\right| \\ v\left(t\right)=\left|\frac{dz}{dt}+i\frac{dy}{dt}\right| \end{gathered}$$

 

Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ as:

 $$\begin{gathered} \frac{dx}{dt}=-2\sin \left(2t\right) \\ \frac{dy}{dt}=2\cos \left(2t\right) \end{gathered}$$

 

Substitute the above values in the formula. The velocity is given below:

 $$\begin{gathered} v\left(t\right)=\sqrt{{\left(2\cos 2t\right)}^2+{\left(2\sin 2t\right)}^2} \\ =\sqrt{4\left[{\left(\cos 2t\right)}^2+{\left(\sin 2t\right)}^2\right]} \\ =2 \end{gathered}$$

 

Step 5: Find the value of the acceleration

The formula for velocity is given below:

 $$\begin{gathered} a\left(t\right)=\left|\frac{d^{2}z}{d{t}^2}\right| \\ =\left|\frac{d^{2}x}{d{t}^2}+\frac{d^{2}y}{d{t}^2}\right| \end{gathered}$$

 

Find $\frac{d^{2}x}{d{t}^2}$ and $\frac{d^{2}y}{d{t}^2}$ as:

 $$\begin{gathered} \frac{d^{2}x}{d{t}^2}=-4\cos \left(2t\right) \\ \frac{d^{2}y}{d{t}^2}=-4\sin \left(2t\right) \end{gathered}$$

 

Substitute the above values in the formula. The acceleration is given below:

 $$\begin{gathered} a\left(t\right)=\sqrt{{\left(4\cos 2t\right)}^2+{\left(4\sin 2t\right)}^2} \\ =\sqrt{16{\left(\cos 2t\right)}^2+{\left(\sin 2t\right)}^2} \\ =4 \end{gathered}$$

 

Step 6: Find the motion

The motion is given below:

 $$\begin{gathered} \left|z\right|=\sqrt{x^2+y^2} \\ \left|z\right|=\sqrt{{\left(\sin 2t\right)}^2+{\left(\cos 2t\right)}^2} \\ \sqrt{x^2+y^2}=1 \\ x^2+y^2=1 \end{gathered}$$

 

Thus, the motion is circular.The value of a and v is given below:

 v = 2,a = 4