Question

A particle moves from the point $\left( {2.0\hat i + 4.0\hat j} \right)m$ at $t = 0$, with an initial velocity $\left( {5.0\hat i + 4.0\hat j} \right)m{s^{ - 1}}$. It is acted upon by a constant force which produces a constant acceleration $\left( {4.0\hat i + 4.0\hat j} \right)m{s^{ - 2}}$. What is the distance of the particle from the origin at time $2s$ ?
A. $20\sqrt 2 m$
B. $10\sqrt 2 m$
C. $5m$
D. $15m$

Answer 1

Hint: To solve the question we will use the second equation of motion in vector form $\vec s = \vec ut + \dfrac{1}{2}\vec a{t^2}$ to and then equate with difference of final and initial position will give the distance and then find magnitude by $|\vec A| = \sqrt {A_x^2 + A_y^2 + A_z^2} $.

Complete step by step answer:
Now from the question
Given initial position vector \[{\vec r_i} = \left( {2.0\hat i + 4.0\hat j} \right)m\]
Initial velocity vector \[\vec u = \left( {5.0\hat i + 4.0} \right)m{s^{ - 1}}\]
Acceleration vector $\vec a = \left( {4.0\hat i + 4.0\hat j} \right)m{s^{ - 2}}$
Now from second equation of motion we have
$\vec s = \vec ut + \dfrac{1}{2}\vec a{t^2}$
Now $\vec s = \left( {5.0\hat i + 4.0\hat j} \right)\left( 2 \right) + \dfrac{1}{2}\left( {4.0\hat i + 4.0\hat j} \right){\left( 2 \right)^2}$
 \[\vec s = \left( {18\hat i + 16\hat j} \right)\]
Now the distance covered by particle from origin at $2s$
\[\begin{gathered}
  {{\vec r}_f} - {{\vec r}_i} = \vec s \\
  {{\vec r}_f} - \left( {2.0\hat i + 4.0\hat j} \right) = \left( {18\hat i + 16\hat j} \right) \\
  {{\vec r}_f} = \left( {18\hat i + 16\hat j} \right) + \left( {2.0\hat i + 4.0\hat j} \right) \\
\end{gathered} \]
${\vec r_f} = \left( {20\hat i + 20\hat j} \right)$

Magnitude of distance:
$|\vec A| = \sqrt {A_x^2 + A_y^2 + A_z^2} $
Hence
$
  |{{\vec r}_f}| = \sqrt {{{\left( {20} \right)}^2} + {{\left( {20} \right)}^2}} \\
  |{{\vec r}_f}| = \sqrt {400 + 400} = \sqrt {800} \\
  |{{\vec r}_f}| = 20\sqrt 2 m \\
 $
Therefore the correct option is (A).

Note:
In such types of questions first note down whatever is given like in the above question we have a point from where a particle moves and its having an initial velocity and after that a constant force acted upon that so to obtain the distance of the particle from the origin we first find out the vector and after that the magnitude of the vector.