Answer

Verified

407.1k+ views

**Hint:**Try to find what is asked in each part by doing proper solving for each and every part separately as per the requirement. Find the average acceleration for part (a), position of the train for part (c) and for part (b) and (d), try to find what is asked accordingly.

**Complete step by step answer:**

a) Average acceleration

$\begin{align}

& =\dfrac{{{\left[ {{v}_{A}}\left( t \right) \right]}_{f}}-{{\left[ {{v}_{A}}\left( t \right) \right]}_{i}}}{{{\left[ t \right]}_{f}}-{{\left[ t \right]}_{i}}} \\

& =\dfrac{{{v}_{A}}\left( 8 \right)-{{v}_{A}}\left( 2 \right)}{8-2} \\

& =\dfrac{-120-100}{6} \\

& =-\dfrac{110}{3}meter/{{\min }^{2}} \\

\end{align}$

b) Yes, as the function ${{v}_{A}}\left( t \right)$is differentiable and thus a continuous function, so the velocity of train A must at some time ‘t’ within 5 < t < 8 equal to -100 meters/min because ${{v}_{A}}$(t)=-100 lies between ${{v}_{A}}$(5)=40 and ${{v}_{A}}$(8)=-120

c)

\[\begin{align}

& x\left( 12 \right)=\int\limits_{2}^{12}{{{v}_{A}}\left( t \right)dt}+x\left( 2 \right) \\

& \Rightarrow x\left( 12 \right)=3.\dfrac{1}{2}.\left( 100+40 \right)+3.\dfrac{1}{2}.\left( 40-120 \right)+4.\dfrac{1}{2}.\left( -120-150 \right)+300 \\

& \Rightarrow x\left( 12 \right)=210-120-540+300 \\

& \Rightarrow x\left( 12 \right)=-150 \\

\end{align}\]

So, at t= 12 minute the train is approximately 150 meters west of origin station

d) Let x be train A’s position, y train B’s position, and z the distance between train A and train B.

\[\begin{align}

& {{x}^{2}}+{{y}^{2}}={{z}^{2}} \\

& \\

\end{align}\]

Differentiating wrt ‘t’;

$\begin{align}

& \Rightarrow 2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}=2z\dfrac{dz}{dt} \\

& Here;x=300,\text{ }y=400 \\

& z=\sqrt{{{300}^{2}}+{{400}^{2}}}=500 \\

& {{v}_{B}}\left( 2 \right)=-20+120+25=125 \\

& \Rightarrow 300\times 100+400\times 125=500\times \dfrac{dz}{dt} \\

& \Rightarrow \dfrac{dz}{dt}=\dfrac{80000}{500}=160meters/\min \\

\end{align}$

**Note:**Calculation should be done according to the requirement of each part of the question. As in part (a) we have to find the average acceleration, in part (c) the position of the train and in part (d) the differentiation.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE