Answer

Verified

385.2k+ views

**Hint:**We need to understand the relation between the acceleration of a body and the displacement of the body. The graph of acceleration versus the time graph can give enough information required for the solution that is needed for this problem.

**Complete step-by-step solution**

We are given a graph of acceleration of a particle versus the time. We know that the slope of certain graphs given the physical parameters related to the quantities on the graph. In our case, the differential of the graph will not give us any required value.

We know that the area under the graph of an acceleration versus time graph can give the velocity of the particle under consideration. We can derive the required relation in terms of time ‘t’ for the velocity using this idea. The integral of the given graph can give the expression for the velocity. So, we can find the equation of the line as –

\[\begin{align}

& y=mx+c \\

& \Rightarrow y=\dfrac{(0-(-2))}{(1-0)}x+-2 \\

& \text{here,} \\

& y=a,x=t \\

& \therefore a=2t-2 \\

\end{align}\]

We can integrate the above equation to get the velocity on the L.H.S as –

\[\begin{align}

& \int\limits_{{}}^{{}}{a}=\int{(2t-2})dt \\

& \Rightarrow v={{t}^{2}}-2t+c \\

& \text{initially,} \\

& v=0 \\

& \therefore v={{t}^{2}}-2t \\

\end{align}\]

This is the required expression for velocity in terms of time ‘t’.

(b) We can find the displacement from the above expression very easily. The displacement is the integral of the velocity with time. So, the displacement of the particle is given as –

\[\begin{align}

& v={{t}^{2}}-2t \\

& \Rightarrow s=\int{v}=\int\limits_{t=2s}^{t=4s}{({{t}^{2}}-2t)}dt \\

& \Rightarrow s=\left[ \dfrac{{{t}^{3}}}{3} \right]_{2}^{4}-[{{t}^{2}}]_{2}^{4} \\

& \therefore s=6.67m \\

\end{align}\]

**The displacement of the particle between the two seconds of motion from 2s to 4s is 6.67m. This is the required solution.**

**Note:**The expression for the acceleration of a particle can give the complete details of the motion of a particle as we have seen in this problem. The same is possible for any of the expressions for the displacement of the particle or the velocity of the particle is given.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

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

Write a letter to the principal requesting him to grant class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail