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Q104P

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Found in: Page 39

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# Question: A proton moves along the axis according to the equation x = 50t + 10t2 , where x is in meters and t is in seconds. Calculate (a) the average velocity of the proton during the first 3.0 s of its motion, (b) the instantaneous velocity of the proton at t = 3.0 s, and (c) the instantaneous acceleration of the proton at t = 3.0 s . (d) Graph x versus t and indicate how the answer to (a) can be obtained from the plot. (e) Indicate the answer to (b) on the graph. (f) Plot v versus t and indicate on it the answer to (c).

1. The average velocity of the proton during the first $3.0\text{s}$ of its motion is $80\text{m/s}$ .
2. The instantaneous velocity of the proton at $t=3.0\text{s}$ is $110\text{m/s}$ .
3. The instantaneous acceleration of the proton at $t=3.0\text{s}$ is $20{\text{m/s}}^{2}$
4. From graph,$x\left(3.0\text{s}\right)=240\text{m,}x\left(0\text{s}\right)=0\text{m}$, Therefore, ${v}_{avg}=80\text{m/s}$
5. The instantaneous velocity at $t=3.0\text{s}$ is $110\text{m/s}$.
6. The instantaneous acceleration from the graph is $20.57{\text{m/s}}^{2}$ .
See the step by step solution

## To understand concept

The equation for the motion of the proton is, $x=50t+10{t}^{2}$

## Understanding the average velocity and instantaneous velocity.

Average velocity is the ratio of total displacement to the total time interval. Instantaneous velocity is the velocity of a moving object at a specific moment.

The expression for the average velocity is given as follows:

${v}_{avg}=\frac{∆x}{∆t}$ … (i)

Here, $∆x$ is the displacement and $∆t$ is the time duration.

The expression for the instantaneous velocity is given as follows:

$v=\frac{dx}{dt}$ … (ii)

The expression for the instantaneous acceleration is given as follows:

$a=\frac{dv}{dt}$ … (iii)

## (a) Determination of the average velocity of the proton during the first 3.0s of its motion

Position of the proton at $t=3.0\text{s}$ is,

localid="1660821352042" $x\left(3.0\text{s}\right)=50\text{m/s×3.0 s+10m/s×}{\left(3.0\text{s}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=150\text{m+90m}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=240\text{m}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Position of the proton at $t=0\text{s}$ is,

$x\left(0\text{s}\right)=50\text{m/s×0 s+10m/s×}{\left(0\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=0\text{m}$

Using equation (i), the average velocity is calculated as follows:

${v}_{avg}=\frac{x\left(3.0\text{s}\right)-x\left(0\text{s}\right)}{{t}_{2}-{t}_{1}}\phantom{\rule{0ex}{0ex}}=\frac{240\text{m-0 m}}{3.0\text{s -0 s}}\phantom{\rule{0ex}{0ex}}=80\text{m/s}$

Thus, the average velocity of the proton is $80\text{m/s}$.

## (b) Determination of the instantaneous velocity of the proton at  t=3.0 s

Using equation (ii), the instantaneous velocity is,

$v=\frac{d\left(50t+10{t}^{2}\right)}{dt}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}v=50+20t$

The instantaneous velocity at $t=3.0\text{s}$ is,

role="math" localid="1650539082653" $v=50+20\left(3.0\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=50+60\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=110\text{m/s}$

Thus, the instantaneous velocity at time$t=3.0\text{s}$ is $110\text{m/s}$ .

## (c) Determination of the instantaneous acceleration of the proton at  t=3.0 s

Using equation (iii), the instantaneous acceleration is,

$a=\frac{d\left(50+20t\right)}{dt}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=20{\text{m/s}}^{2}$

Thus, the instantaneous acceleration at $t=3.0\text{s}$ is $20{\text{m/s}}^{2}$ .

## (d) Determination of average velocity from the graph.

The graph x vs t is plotted below.

From the graph,

The positions at 3.0 s and 0 s are,

$x\left(3.0\text{s}\right)=240\text{m and x}\left(0\text{s}\right)\text{=0 m}$

So, the average velocity is,

${v}_{avg}=\frac{240\text{m}}{3.0\text{s}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=80\text{m/s}$

## (e) Indicate the instantaneous velocity of the proton at t=3.0 s  on the plot

The instantaneous velocity at $t=3.0\text{s}$ is the slope of the triangle drawn at that point in the above graph and it is,

$\text{slope=}\frac{350-120}{4-1.9}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\approx 110\text{m/s}$

Thus, the instantaneous velocity at 3.0 s is 110 m/s.

## (f) Plot the graph of V vs t

The graph of $v$ vs $t$ is as below,

From the graph, the instantaneous acceleration is the slope of the graph

$\text{Slope =}\frac{132-60}{4-0.5}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=20.57{\text{m/s}}^{2}$

Thus, the instantaneous acceleration from the graph is$20.57{\text{m/s}}^{2}$ .