Question

You are given the equations that are used to solve a problem. For each of these, you are to

a. Write a realistic problem for which these are the correct equations. Be sure that the answer your problem requests is consistent with the equations given.

b. Finish the solution of the problem, including a pictorial representation.

$$\begin{gathered} 2.5\mathrm{rad}=0\mathrm{rad}+{\mathrm{\omega }}_{\mathrm{i}}(10\mathrm{s})+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/2(50\mathrm{m}\left)\right){(10\mathrm{s})}^2 \\ {\mathrm{\omega }}_{\mathrm{f}}={\mathrm{\omega }}_{\mathrm{i}}+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/(50\mathrm{m}\left)\right)(10\mathrm{s}) \end{gathered}$$

Short answer

$$\begin{gathered} {\mathrm{\omega }}_{\mathrm{i}}=0.1\mathrm{rad}/\mathrm{s} \\ {\mathrm{\omega }}_{\mathrm{f}}=0.4\mathrm{rad}/\mathrm{s} \end{gathered}$$

Step-by-step solution

Given information

We are given with the equations:

$$\begin{gathered} 2.5\mathrm{rad}=0\mathrm{rad}+{\mathrm{\omega }}_{\mathrm{i}}(10\mathrm{s})+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/2(50\mathrm{m}\left)\right){(10\mathrm{s})}^2 \\ {\mathrm{\omega }}_{\mathrm{f}}={\mathrm{\omega }}_{\mathrm{i}}+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/(50\mathrm{m}\left)\right)(10\mathrm{s}) \end{gathered}$$

a. We have to write a realistic problem for which these are the correct equations.

b. Finish the solution of the problem, including a pictorial representation.

Part a) Step 1: Realistic Problem 

A car tire is rotating at $2.5\mathrm{rad}$. After traveling $50\mathrm{m}$in $10\mathrm{s}$with the acceleration of $1.5\mathrm{m}/{\mathrm{s}}^2$.What is the car tire's initial and final velocity?

Part b) Step 1: Calculation

We are given with two equations, from first equation:

$$\begin{gathered} 2.5\mathrm{rad}=0\mathrm{rad}+{\mathrm{\omega }}_{\mathrm{i}}(10\mathrm{s})+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/2(50\mathrm{m}\left)\right){(10\mathrm{s})}^2 \\ 2.5\mathrm{rad}={\mathrm{\omega }}_{\mathrm{i}}(10\mathrm{s})+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/2(50\mathrm{m}\left)\right)100{\mathrm{s}}^2 \\ 2.5\mathrm{rad}={\mathrm{\omega }}_{\mathrm{i}}(10\mathrm{s})+1.5 \\ 1\mathrm{rad}=\mathrm{\omega i}(10\mathrm{s}) \\ {\mathrm{\omega }}_{\mathrm{i}}=0.1\mathrm{s} \end{gathered}$$

From second equation:

$$\begin{gathered} {\mathrm{\omega }}_{\mathrm{f}}={\mathrm{\omega }}_{\mathrm{i}}+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/(50\mathrm{m}\left)\right)(10\mathrm{s}) \\ {\mathrm{\omega }}_{\mathrm{f}}=0.1\mathrm{rad}/\mathrm{s}+\left(\right(1.5\mathrm{m}/{\mathrm{s}}^2)/(50\mathrm{m}\left)\right)(10\mathrm{s}) \\ {\mathrm{\omega }}_{\mathrm{f}}=0.1\mathrm{rad}/\mathrm{s}+0.3 \\ {\mathrm{\omega }}_{\mathrm{f}}=0.4\mathrm{rad}/\mathrm{s} \end{gathered}$$

Part b) Step 2: Pictorial representation

The Pictorial representation of the given equations are as: