Chapter 2, Problem 66PE

To determine

The graph of velocity and acceleration of the particle.

Answer to Problem 66PE

The graph of velocity and acceleration of the particle is shown in figure (2) and figure (3) respectively.

Explanation of Solution

Given:

The given figure is,

  

  Figure (1)

Formula used:

The velocity of particle is given as,

  $v=\frac{\Delta x}{\Delta t}$

The acceleration of particle is given as,

  $a=\frac{v_2-v_1}{t_2-t_1}$

Calculation:

From figure (1),

Position (m)	Time (s)
$0$	$0$
$1$	$1$
$2$	$2$
$3$	$-3$
$4$	$-3$
$5$	$-3$
$6$	$-2$

The velocity of particle from $0\text{s}$ to $1\text{s}$ time interval is calculated as,

  $\begin{array}{c}{v}_1=\frac{\Delta x}{\Delta t}\\ =\frac{1\text{m}-0\text{m}}{1\text{s}-0\text{s}}\\ =1\text{m}/\text{s}\end{array}$

The velocity of particle from $1\text{s}$ to $2\text{s}$ time interval is calculated as,

  $\begin{array}{c}{v}_2=\frac{\Delta x}{\Delta t}\\ =\frac{2\text{m}-1\text{m}}{2\text{s}-1\text{s}}\\ =1\text{m}/\text{s}\end{array}$

The velocity of particle from $2\text{s}$ to $3\text{s}$ time interval is calculated as,

  $\begin{array}{c}{v}_3=\frac{\Delta x}{\Delta t}\\ =\frac{-3\text{m}-2\text{m}}{3\text{s}-2\text{s}}\\ =-5\text{m}/\text{s}\end{array}$

The velocity of particle from $3\text{s}$ to $4\text{s}$ time interval is calculated as,

  $\begin{array}{c}{v}_4=\frac{\Delta x}{\Delta t}\\ =\frac{-3\text{m}-\left(-3\text{m}\right)}{4\text{s}-3\text{s}}\\ =0\text{m}/\text{s}\end{array}$

The velocity of particle from $4\text{s}$ to $5\text{s}$ time interval is calculated as,

  $\begin{array}{c}{v}_5=\frac{\Delta x}{\Delta t}\\ =\frac{-3\text{m}-\left(-3\text{m}\right)}{5\text{s}-4\text{s}}\\ =0\text{m}/\text{s}\end{array}$

The velocity of particle from $5\text{s}$ to $6\text{s}$ time interval is calculated as,

  $\begin{array}{c}{v}_6=\frac{\Delta x}{\Delta t}\\ =\frac{-2\text{m}-\left(-3\text{m}\right)}{6\text{s}-5\text{s}}\\ =1\text{m}/\text{s}\end{array}$

The graph of velocity versus time of the particle is shown below.

  

  Figure (2)

The acceleration of particle is calculated as,

  $\begin{array}{c}a=\frac{2\text{m}/\text{s}-2\text{m}/\text{s}}{2\text{s}-2\text{s}}\\ =0{\text{m}/\text{s}}^2\end{array}$

From figure (2) it can be seen that the velocity suddenly changes at $2\text{s}$, $3\text{s}$, $4\text{s}$ and $5\text{s}$ so there is some acceleration spikes at those points.

The graph of acceleration versus time of the particle is shown below.

  

  Figure (3)

Conclusion:

The graph of velocity and acceleration of the particle is shown in figure (2) and figure (3) respectively.