Question

Emmy is standing on a moving sidewalk that moves at +2 m/s. Suddenly, she realizes she might miss her flight, so begins to speed up, but then she hears over the PA system that her flight is delayed.

The acceleration versus time graph shows Emily's acceleration during the 5s time interval in which all of the events occur. In the velocity versus time graph, construct a plot of Emmy's velocity over this same time period.

Assuming all the values given are exact, what is Emmy's velocity at 3.53s?

Acceleration (m/s) B

Answer 1

$\begin{array}{l}initial\;velocity\;of\;emmy,\;i.e.\;velocity\;at\;time\;t=0,\;is\;zero.\\v_0=0\;m/s\\step:1\\now\;fot\;time\;t=0\;\;to\;\;\;t=1\;\\acceleration\;of\;emmy\;is\;,\;a=\;2\;m/s^2\\so\;u\sin g\;equation\;\\v_1=v_0+at\\v_1=0+(2\times1)\\v_1=\;2\;m/s\;\;\;\rightarrow(1)\;\;\;\;\;where\;v_1\rightarrow velocity\;after\;1\;second\\\end{array}$

$\begin{array}{l}step:2\\for\;time\;t=1\;to\;t=2\;\;\;\;i.e.\;time\;interval\;\triangle t=\;(2-1)=1\;s\\acceleration,\;a=1\;m/s^2\\the\;velocity\;we\;obtained\;in\;step\;1\;will\;act\;here\;as\;initial\;velocity.\\thus\;velocity\;after\;2\;second\;will\;be\\v_2=v_1+a(\triangle t)\\v_2=2+(1\times1)\\v_2=3\;m/s\;\;\rightarrow(2)\\where\;v_2\rightarrow velocity\;after\;t=2\;seconds\end{array}$

$\begin{array}{l}step:3\\for\;time\;t=2\;to\;t=3\;\;\;i.e.\;time\;interval\;\triangle t=\;(3-2)=1\;s\\acceleration,\;a=0\;m/s^2\\the\;velocity\;we\;obtained\;in\;step\;2\;will\;act\;here\;as\;initial\;velocity.\\thus\;velocity\;after\;3\;second\;will\;be\\v_3=v_2+a(\triangle t)\\v_3=3+(0\times1)\\v_3=3\;m/s\;\;\rightarrow(2)\\where\;v_3\rightarrow velocity\;after\;t=3\;seconds\end{array}$

$\begin{array}{l}step:4\\for\;time\;t=3\;to\;t=4\;\;\;i.e.\;time\;interval\;\triangle t=\;(4-3)=1\;s\\acceleration,\;a=-3\;m/s^2\\the\;velocity\;we\;obtained\;in\;step\;3\;will\;act\;here\;as\;initial\;velocity.\\thus\;velocity\;after\;4\;second\;will\;be\\v_4=v_3+a(\triangle t)\\v_4=3+(-3\times1)\\v_4=0\;\;m/s\;\;\rightarrow(2)\\where\;v_4\rightarrow velocity\;after\;t=4\;seconds\end{array}$

$\begin{array}{l}step:5\\for\;time\;t=4\;to\;t=5\;\;\;i.e.\;time\;interval\;\triangle t=\;(5-4)=1\;s\\acceleration,\;a=0\;m/s^2\\the\;velocity\;we\;obtained\;in\;step\;4\;will\;act\;here\;as\;initial\;velocity.\\thus\;velocity\;after\;5\;second\;will\;be\\v_5=v_4+a(\triangle t)\\v_5=0+(0\times1)\\v_5=0\;m/s\;\;\rightarrow(2)\\where\;v_5\rightarrow velocity\;after\;t=5\;seconds\end{array}$

so if we plot all the obtained velocities v₁ ,v₂ ,v₃ ,v₄ ,v₅ with time, the plot will be following

vocity m/a) 3 2 time 3 - 2 +ime Plot FigVelocity

$\begin{array}{l}at\;t=3\;emmy's\;velocity\;is\\v_3=3\;m/s\\acceleration\;between\;t=3\;to\;t=\;4,\;a=-3\;m/s^2\\thus\;velocity\;at\;t=3.53\;s\;\;\;\;\;\;\;\;time\;interval\;\;\triangle t=(3.53-3.00)=0.53\;s\\v=v_3\;+a(\triangle t)\\v=3+(-3\times0.53)\\v=3-1.59\\v=1.41\;m/s\;\;\;\;\;(answer)\end{array}$