A 4 kg particle moves under the force $[ \vec F = 3(4t\hat i - 3t\hat j)\ \text{N} ]$ If the particle starts from origin at (t = 0), find its velocity and position at t = 2 s.

Options:

1. $( \vec v = \frac{3}{2}\hat i + \frac{6}{5}\hat j,\ \vec r = \hat i + \hat j )$
2. $( \vec v = \frac{3}{2}\hat i - \frac{8}{5}\hat j,\ \vec r = \hat i - \hat j )$
3. $( \vec v = \frac{5}{2}\hat i + \frac{8}{5}\hat j,\ \vec r = \hat i + 2\hat j )$
4. $( \vec v = \frac{3}{2}\hat i - \frac{6}{5}\hat j,\ \vec r = \hat i - \hat j )$

Solution:

Using (F = ma), integrate acceleration → velocity → position. Final values at (t = 2s): $[ \vec v = \frac{3}{2}\hat i - \frac{8}{5}\hat j ]$ $[ \vec r = \hat i - \hat j ]$ ✅ Correct Answer: (2)