Question

A body of mass 2 kg moving with velocity of $\vec{v_{in}}$ = $3\hat{i}$ + $4\hat{j}$ $ms^{-1}$ enters into a constant force field of 6N directed along positive z-axis. If the body remains in the field for a period of $\frac{5}{3}$ seconds, then velocity of the body when it emerges from force field is.

• A. $3\hat{i}$ + $4\hat{j}$ + $\sqrt{5}\hat{k}$
• B. $4\hat{i}$ + $3\hat{j}$ + $5\hat{k}$
• C. $3\hat{i}$ + $4\hat{j}$ - $5\hat{k}$
• D. $3\hat{i}$ + $4\hat{j}$ + $5\hat{k}$

Answer 1

Answer: (D)

$3\hat{i} + 4\hat{j} + 5\hat{k}$

Answer 2

The problem involves calculating the final velocity of a body subjected to a constant force for a given duration.

1. Identify the given parameters:

• Mass of the body, $m = 2$ kg
• Initial velocity of the body, $\vec{v_{in}} = 3\hat{i} + 4\hat{j}$ $ms^{-1}$
• Constant force acting on the body, $\vec{F} = 6$ N along the positive z-axis. So, $\vec{F} = 6\hat{k}$ N
• Time duration for which the force acts, $t = \frac{5}{3}$ seconds

2. Calculate the acceleration of the body:

According to Newton's second law, $\vec{F} = m\vec{a}$. Therefore, the acceleration $\vec{a} = \frac{\vec{F}}{m}$. $\vec{a} = \frac{6\hat{k}}{2} = 3\hat{k}$ $ms^{-2}$

3. Calculate the final velocity of the body:

Since the acceleration is constant, we can use the kinematic equation: $\vec{v_{final}} = \vec{v_{initial}} + \vec{a}t$. Substitute the values: $\vec{v_{final}} = (3\hat{i} + 4\hat{j}) + (3\hat{k}) \times \left(\frac{5}{3}\right)$ $\vec{v_{final}} = 3\hat{i} + 4\hat{j} + \left(\frac{3 \times 5}{3}\right)\hat{k}$ $\vec{v_{final}} = 3\hat{i} + 4\hat{j} + 5\hat{k}$ $ms^{-1}$

The x and y components of the velocity remain unchanged because the force acts only along the z-axis, meaning there is no acceleration component in the x or y directions. Only the z-component of the velocity changes due to the applied force.

The final velocity of the body when it emerges from the force field is $3\hat{i} + 4\hat{j} + 5\hat{k}$ $ms^{-1}$.