A body of mass 5 kg under the action of constant force $$\vec{F} = F_x\hat{i} + F_y\hat{j}$$ has velocity at t = 0 s as $$\vec{v} = (6\hat{i} - 2\hat{j})$$ m/s and at t = 10 s as $$\vec{v} = +6\hat{j}$$ m/s. The force $$\vec{F}$$ is:

We are given the mass of the body, $$ m = 5 $$ kg, the initial velocity at $$ t = 0 $$ s as $$ \vec{u} = (6\hat{i} - 2\hat{j}) $$ m/s, and the final velocity at $$ t = 10 $$ s as $$ \vec{v} = +6\hat{j} $$ m/s. The force $$ \vec{F} = F_x\hat{i} + F_y\hat{j} $$ is constant. We need to find $$ \vec{F} $$.

According to Newton's second law, the force is given by $$ \vec{F} = m\vec{a} $$, where $$ \vec{a} $$ is the acceleration. Acceleration is the rate of change of velocity, so $$ \vec{a} = \frac{\Delta \vec{v}}{\Delta t} $$.

First, calculate the change in velocity $$ \Delta \vec{v} $$:

$$ \Delta \vec{v} = \vec{v} - \vec{u} = (6\hat{j}) - (6\hat{i} - 2\hat{j}) $$

Distribute the negative sign:

$$ \Delta \vec{v} = 6\hat{j} - 6\hat{i} + 2\hat{j} $$

Combine like terms:

$$ \Delta \vec{v} = -6\hat{i} + (6 + 2)\hat{j} = -6\hat{i} + 8\hat{j} \text{ m/s} $$

The time interval $$ \Delta t = t_{\text{final}} - t_{\text{initial}} = 10 - 0 = 10 $$ s.

Now, find the acceleration $$ \vec{a} $$:

$$ \vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{-6\hat{i} + 8\hat{j}}{10} $$

Separate the components:

$$ \vec{a} = \left( \frac{-6}{10} \right) \hat{i} + \left( \frac{8}{10} \right) \hat{j} $$

Simplify the fractions:

$$ \vec{a} = -\frac{6}{10}\hat{i} + \frac{8}{10}\hat{j} = -\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \text{ m/s}^2 $$

Now, use Newton's second law to find the force:

$$ \vec{F} = m \vec{a} = 5 \times \left( -\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \right) $$

Distribute the mass to each component:

$$ \vec{F} = 5 \times \left( -\frac{3}{5} \right) \hat{i} + 5 \times \left( \frac{4}{5} \right) \hat{j} $$

Simplify each term:

$$ \vec{F} = \left( 5 \times -\frac{3}{5} \right) \hat{i} + \left( 5 \times \frac{4}{5} \right) \hat{j} $$

$$ \vec{F} = \left( -3 \right) \hat{i} + \left( 4 \right) \hat{j} $$

So, $$ \vec{F} = (-3\hat{i} + 4\hat{j}) $$ N.

Comparing with the options:

• A. $$ (-3\hat{i} + 4\hat{j}) $$ N
• B. $$ \left( -\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \right) $$ N
• C. $$ (3\hat{i} - 4\hat{j}) $$ N
• D. $$ \left( \frac{3}{5}\hat{i} - \frac{4}{5}\hat{j} \right) $$ N

Our result matches option A.

Hence, the correct answer is Option A.