Question

An object starts from rest, and moves under the acceleration $a=4i$ its position after 3s is given by $r=7i+4j$ what is its initial position?

Answer 1

Hint: First of all we will use a displacement formula in which we will substitute value of displacement $\left( \overrightarrow{S} \right)$ with $\overrightarrow{S}=$ final position – initial position. After doing that we can find the initial position of an object.
Formula used: $\overrightarrow{S}=\overrightarrow{u}t+\dfrac{1}{2}\overrightarrow{a}{{t}^{2}}$

Complete answer:
Given data
$\begin{align}
  & \overrightarrow{a}=4\widehat{i} \\
 & t=3s \\
 & \overrightarrow{r}=7\widehat{i}+4\widehat{j} \\
\end{align}$
Here $\overrightarrow{r}$ is the final position so let’s assume initial position is$\overrightarrow{q}$,
Now displacement $\left( \overrightarrow{S} \right)$
$\overrightarrow{S}=$ Final position – initial position
 $\overrightarrow{S}=\overrightarrow{r}-\overrightarrow{q}.....(1)$
Now formula for displacement as equation of motion,
$\overrightarrow{S}=\overrightarrow{u}t+\dfrac{1}{2}\overrightarrow{a}{{t}^{2}}.....\left( 2 \right)$
Where, $\overrightarrow{S}=$ displacement of the particle
$\overrightarrow{u}$= initial velocity of the particle
t = time
$\overrightarrow{a}$ = acceleration of the particle
Here the object start moving from the rest hence its initial velocity is zero $\overrightarrow{u}=0$
Now substitute the value of equation (1) into equation (2)
$\begin{align}
  & \Rightarrow \overrightarrow{r}-\overrightarrow{q}=\left( 0 \right)\times \left( 3 \right)+\dfrac{1}{2}(4\widehat{i}){{(3)}^{2}} \\
 & \Rightarrow 7\widehat{i}+4\widehat{j}-\overrightarrow{q}=\dfrac{1}{2}\times 36\widehat{i} \\
 & \Rightarrow -\overrightarrow{q}=18\widehat{i}-7\widehat{i}-4\widehat{j} \\
 & \Rightarrow -\overrightarrow{q}=+11\widehat{i}-4\widehat{j} \\
 & \therefore \overrightarrow{q}=-11\widehat{i}+4\widehat{j} \\
\end{align}$
 Therefore initial position of particle is $\overrightarrow{q}=-11\widehat{i}+4\widehat{j}$

Additional information:
Newton’s first law of motion states that if the body is at rest or moving at the constant speed in a straight line it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by force. For example in space when we put some object and push it by some force it will go in the direction of force with constant speed until and unless any other force acts on it.
Newton's second law states that the acceleration of an object is directly related to the net force and inversely related to its mass. Acceleration of an object depends on two things, force and mass.

Note:
In this question we need to remember two things that when object is at rest is initial
velocity is zero and after putting displacement use the formula as equation of motion we should know basic of displacement that is equal to (final position – initial position) as we want to find initial position of object.