Q90P - Fundamentals Of Physics

Question

A particle starts from the origin at $\mathit{t}\mathbf{=}\mathbf{0}$ and moves along the positive $\mathit{x}\mathbf{ }\mathit{a}\mathit{x}\mathit{i}\mathit{s}$. A graph of the velocity of the particle as a function of the time is shown in Fig. 2-46; the $\mathit{v}\mathbf{-}\mathit{a}\mathit{x}\mathit{i}\mathit{s}$ scale is set by vs $\mathbf{4}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$. (a) What is the coordinate of the particle at $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (b) What is the velocity of the particle at$\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (c) What the acceleration of the particle at $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (d) What is the average velocity of the particle between $\mathit{t}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathit{s}$ and $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (e) What is the average acceleration of the particle between $\mathit{t}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathit{s}$ and $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$?

Step-by-Step Solution

Answer

1. The particle is at $x=15\mathrm{m}$ at $5.00\mathrm{s}$
2. Velocity of the particle at $5.00\mathrm{s}$ is $2 \mathrm{m}/\mathrm{s}$
3. Acceleration of the particle at $5.00sec$ is $-2 \mathrm{m}/{\mathrm{s}}^2$
4. Average velocity between $t=1 s$ to $t=5 s$ is $3.5 \mathrm{m}/\mathrm{s}$
5. Average acceleration of the particle between $t=1 s$ to $t=5 s$ is $0 \mathrm{m}/{\mathrm{s}}^2$

Step 1: To understand the concept

The problem is based on the simple concept of calculating area of triangle. The calculation of region bounded by the curve within the given graph is called as area under the curve.

 

Formula:

 

The area of triangle is given by,

$A=\frac{1}{2}\left(\mathrm{base}\times \mathrm{height}\right)$

Step 2 (a): Determination of position of particle at t = 5 . 0   s

To find the coordinates at $5 \sec$, we need to know the displacement of the particle at $5 \sec$. The displacement of the particle would be equal to the area under the plot at $t = 5 \sec$.

Area of the triangle for $t=2 \sec$ is

$\begin{array}{c}{A}_1=\frac{1}{2}\times 2\times 4\\ =4 \mathrm{m}\end{array}$

Area of the rectangle for $t=2 \sec \mathrm{to} t=4 \sec$ is 

$\begin{array}{c}{A}_2=2\times 4\\ =8\mathrm{m}\end{array}$

Area of the shape for $t=4 \sec \mathrm{to} t=5 \sec$

$\begin{array}{c}{A}_3=\left(\frac{1}{2}\times 1\times 2\right)+\left(1\times 2\right)\\ =3 \mathrm{m}\end{array}$

So, the total displacement is 

$\left(A_1+A_2+A_3\right)$

$4 \mathrm{m}+8 \mathrm{m}+3 \mathrm{m}=15 \mathrm{m}$

So, the coordinates at $\mathrm{t}=5 \sec$ are $\left(5, 15\right)$.

Step 3 (b): Determination of velocity on of particle at t = 5 . 0   s

Velocity at  $5.00 \sec$ is given from graph as $2 \mathrm{m}/\mathrm{s}$.

Step 4 (c): Determination of acceleration of particle at t = 5 . 0   s

Acceleration at $5.0 \sec$ can be found by finding the slope of the graph at $\mathrm{t}=5.0 \sec$. It is equal to $\mathrm{a}=-2 \mathrm{m}/{\mathrm{s}}^2$. It is negative since the velocity is reducing.

Step5 (d): Determination of average velocity of particle between t = 1 . 0   s   a n d   t = 5 . 0   s

Using the method used in (a), we can find the displacement at $\mathrm{t}=1 \sec$, which is equal to $1 \mathrm{m}$, we already know the displacement at $\mathrm{t}=5 \sec$ which is $15 \mathrm{m}$

$\begin{array}{c}{v}_{avg}=\frac{\Delta x}{\Delta t}\\ =\frac{15-1}{5-1}\\ =\frac{14}{4}\\ =3.5 \mathrm{m}/\mathrm{s}\end{array}$

So, the average velocity would be $3.5 \mathrm{m}/\mathrm{s}$.

Step 6 (e): Determination of average acceleration of particle between t = 1 . 0   s   a n d   t = 5 . 0   s

The average acceleration at $1.0 \sec$ and $5.0 \sec$

Velocity at $t = 1 \sec$ is $2 \mathrm{m}/\mathrm{s}$ and velocity at $t=5 \sec$ is $2 \mathrm{m}/\mathrm{s}$

$\begin{array}{c}{\mathrm{a}}_{\mathrm{avg}}=\frac{\Delta v}{\Delta t}\\ =\frac{2-2}{4}\\ =0\end{array}$

So, the average acceleration would be $0$.