Q5P - Mathematical Methods in Physical Sciences

Question

In each of the following problems, z represents the displacement of a particle from the origin. Find (as functions of t) its speed and the magnitude of its acceleration, and describe the motion.

$\mathit{z}\mathbf{=}{\mathit{z}}_{\mathbf{1}}\mathit{t}\mathbf{+}{\mathit{z}}_{\mathbf{2}}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{t}\mathbf{)}$Hint: See Problem 4; the straight line here is through the points ${\mathit{z}}_{\mathbf{1}}$ and ${\mathit{z}}_{\mathbf{2}}$

Step-by-Step Solution

Answer

The part it follows is straight line.

Its speed is $z_1-z_2$.

Acceleration $0$

$z_i=z_2 \mathrm{and} z_f=z_2$

Step 1: Given Information.

The given expression is $z=z_{1}t+z_2(1-t)$.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ in which x is the real part and y is the imaginary part.

Step 3: Simplify.

Rewrite the given equation.

$$\begin{gathered} z\left(t\right)=z_1 t-z_{2}t+z_2 \\ z\left(t\right)=\left[z_1-z_2\right]t+z_2 \end{gathered}$$

 

It is of the form $z\left(t\right)=a t+b$

Therefore, it represents a straight line.

Step 4: Find the velocity and acceleration.

Differentiate with respect to time.

$$\begin{gathered} v\left(t\right)=\frac{dz\left(t\right)}{dt} \\ v\left(t\right)=\frac{d}{dt}\left[\left(z_{1-}{z}_2\right)t+z_2\right] \\ v\left(t\right)=z_1-z_2 \end{gathered}$$

 

 

Find the acceleration.

Differentiate the velocity with respect to time.

$\begin{array}{l}A\left(t\right)=\frac{dv\left(t\right)}{dt}\\ A\left(t\right)=0\end{array}$

Step 5: Find the initial and final position.

Find $z\left(0\right)$ for the initial position.

$z\left(0\right)=z_2$

 

Find Final position.

$z_f=z_1$

 

Hence the path it follows is straight line.

Its speed is $z_1=z_2$.

Acceleration 0

$$\begin{gathered} z_i=z_2 \mathrm{and} \\ {z}_f=z_1 \end{gathered}$$