An object with mass \( 5 \mathrm{~kg} \) is acted upon by a force, \( \vec{F}=(-3 \hat{i}+4 \hat{j}) \mathrm{N} \). If its initial velocity at
\( \mathrm{t}=0 \) is \( \vec{v}=(6 \hat{i}-12 \hat{j}) \mathrm{ms}^{-1} \), the time at which it will just have a velocity along \( \mathrm{y} \)-axis is

(D)
\(\vec{F}=(-3 \hat{i}+4 \hat{j})\)
\(m \vec{a}=(-3 \hat{i}+4 \hat{j})\)
\(M=5 \mathrm{~kg}\)
\(\vec{a}=\left(\frac{-3}{5} \hat{i}+\frac{4}{5} \hat{j}\right) \Rightarrow \vec{a}=a_{x} \hat{i}+a_{y} \hat{j}\)
\(a_{x}=\frac{-3}{5}\)
at \(\mathrm{t}=0\)
\(\vec{V}_{0}=(6 \hat{i}-12 \hat{j}) \Rightarrow\left(V_{o x} \hat{i}-V_{o y} \hat{j}\right)\)
\(V_{o x}=6\)
For the body to have velocity along y-axis only, x-component of velocity should be zero i.e., \(V_{x}=0\)
\(V_{x}=V_{o x}+a_{x} t\)
\(0=6+\left(\frac{-3}{5}\right) t\)
\(6=\frac{3}{5} t\)
\(t=\frac{6 \times 5}{3}=10 s\)
OR
\(A=\frac{F}{m}=\left(-\frac{3}{5} \hat{i}+\frac{4}{5} \hat{j}\right)\)
\(\mathrm{u}=(6 \hat{i}-12 \hat{j}) \mathrm{m} s^{-1}\)
\(\mathrm{v}=\mathrm{u}+\) at.
\(1 \hat{j}=(6 \hat{i}-12 \hat{j})+\left(-\frac{3}{5} \hat{i}+\frac{4}{5} \hat{j}\right) t\)