Question

At any time t, a particle of mass 0.1kg has position vector, \[\vec{r}=\left( 5t\hat{i}+4\cos t\hat{j} \right)m\].The average acceleration during the time \[0\le t\le \dfrac{\pi }{2}\] is:
\[\begin{align}
  & A)\dfrac{4}{\pi }m/{{s}^{2}} \\
 & B)\dfrac{\pi }{4}m/{{s}^{2}} \\
 & C)\dfrac{8}{\pi }m/{{s}^{2}} \\
 & D)\dfrac{\pi }{8}m/{{s}^{2}} \\
\end{align}\]

Answer 1

Hint: Position vector of the particle is given in the question. We know that velocity of an object is the rate of change of displacement and average acceleration is the change in velocity during a time interval. Using this relation, we can find out the velocity and thereby average acceleration of the particle.

Formula used:
\[v=\dfrac{dr}{dt}\]
\[{{\vec{a}}_{avg}}=\dfrac{{{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}}}{\Delta t}\]

Complete step by step solution:
Given,
Position of the particle, \[\vec{r}=\left( 5t\hat{i}+4\cos t\hat{j} \right)m\]
We have,
\[v=\dfrac{dr}{dt}\]
Where,
\[v\] is the velocity of the particle
\[\dfrac{dr}{dt}\] is the rate of change of position.
Then,
\[\vec{v}=\dfrac{d\vec{r}}{dt}=\dfrac{d\left( 5t\hat{i}+4\cos t\hat{j} \right)}{dt}=\left( 5\hat{i}-4\sin t\hat{j} \right)m/s\]
Average acceleration, \[{{\vec{a}}_{avg}}=\dfrac{{{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}}}{\Delta t}\]
Where,
\[{{\vec{v}}_{1}}\] is the velocity of the particle at time \[t=\dfrac{\pi }{2}\]
\[{{\vec{v}}_{2}}\] is the velocity of the particle at time \[t=0\]
\[\Delta t\] is the change in time
Therefore,
Average acceleration during the time\[0\le t\le \dfrac{\pi }{2}\] is,
\[\begin{align}
  & {{{\vec{a}}}_{avg}}=\dfrac{\left( 5\hat{i}-4\sin \dfrac{\pi }{2}\hat{j} \right)-\left( 5\hat{i}-4\sin 0\hat{j} \right)}{\dfrac{\pi }{2}-0}=\dfrac{-4\hat{j}\times 2}{\pi }=\dfrac{-8\hat{j}}{\pi }m/{{s}^{2}} \\
 & \\
\end{align}\]
\[{{\vec{a}}_{avg}}=\dfrac{8}{\pi }\left( -\hat{j} \right)m/{{s}^{2}}\]

Therefore, the answer is option C.

Additional information:
When the velocity of an object changes, we say the object is accelerating. Thus, acceleration of an object is the rate of change of its velocity with time. Like velocity, acceleration is also represented with its magnitude and direction. It includes changes in speed as well as changes in direction of the object. Hence, the object is said to be accelerating, when either the direction or the velocity is changing.
According to Newton’s law, mass times acceleration is known as force. Hence, the acceleration of the object depends on the mass of the object and the external force applied to it. Acceleration of the object decreases with increased mass, and increases if the force applied is increased.

Note:
There are two types of acceleration: average and instantaneous. Average acceleration of the object is determined over a long interval of time. That is, for a finite period of time. Instantaneous acceleration is the acceleration of the object at a particular instant.