A force $\overrightarrow{F}$ = $2\hat{i} + 3\hat{j} + 6\hat{k}$ Newton acts on a 5 kg object for 10 second, then the change in velocity of object will be:

Question

Accepted Answer

To determine the change in velocity of the object, we can use Newton's second law of motion, which states that the force acting on an object is equal to the rate of change of its momentum. For a constant mass, this can be expressed as:

$$
\overrightarrow{F} = m \cdot \overrightarrow{a}
$$

where $\overrightarrow{F}$ is the force vector, $m$ is the mass of the object, and $\overrightarrow{a}$ is the acceleration vector. The acceleration can be found by rearranging the equation:

$$
\overrightarrow{a} = \frac{\overrightarrow{F}}{m}
$$

Given:
- $\overrightarrow{F} = 2\hat{i} + 3\hat{j} + 6\hat{k}$ Newton
- $m = 5$ kg

Substituting the given values:

$$
\overrightarrow{a} = \frac{2\hat{i} + 3\hat{j} + 6\hat{k}}{5} = \frac{2}{5}\hat{i} + \frac{3}{5}\hat{j} + \frac{6}{5}\hat{k}
$$

The change in velocity $\Delta \overrightarrow{v}$ can be calculated using the formula:

$$
\Delta \overrightarrow{v} = \overrightarrow{a} \cdot t
$$

where $t = 10$ seconds. Substituting the values:

$$
\Delta \overrightarrow{v} = \left(\frac{2}{5}\hat{i} + \frac{3}{5}\hat{j} + \frac{6}{5}\hat{k}\right) \cdot 10 = 4\hat{i} + 6\hat{j} + 12\hat{k}
$$

The magnitude of the change in velocity is given by:

$$
|\Delta \overrightarrow{v}| = \sqrt{(4)^2 + (6)^2 + (12)^2} = \sqrt{16 + 36 + 144} = \sqrt{196} = 14 \text{ m/s}
$$

Thus, the change in velocity of the object is 14 m/s, which corresponds to Option 2.

**Explanation for Incorrect Options:**

1. **Option 1 (12 m/s):** This option is incorrect because the calculation of the magnitude of the change in velocity results in 14 m/s, not 12 m/s. The components of the velocity change vector do not sum to a magnitude of 12 m/s.

3. **Option 3 (24 m/s):** This option is incorrect because the magnitude of the change in velocity is calculated as 14 m/s. The value of 24 m/s does not match the result of the vector magnitude calculation.

4. **Option 4 (36 m/s):** This option is incorrect because the magnitude of the change in velocity is 14 m/s. The value of 36 m/s is significantly higher than the calculated magnitude and does not correspond to the correct result.

Therefore, the correct answer is indeed Option 2, with a change in velocity of 14 m/s.