The force on a particle of mass `10g` is `(hati 10+hatj 5)`N If it starts from rest what would be its position at time `t=5s` ?

Options

A. `12500 hati+6250 hatj m`
B. `6250 hati+12500 hatj m`
C. `12500 hati+12500 hatj m`
D. `6250 hati+6250 hatj m`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given data
- Mass of the particle, \( m = 10 \, \text{g} = 0.01 \, \text{kg} \)
- Force acting on the particle, \( \vec{F} = 10 \hat{i} + 5 \hat{j} \, \text{N} \)
- Time, \( t = 5 \, \text{s} \)

### Step 2: Calculate the acceleration
Using Newton's second law, we can find the acceleration \( \vec{a} \) using the formula:
\[ \vec{a} = \frac{\vec{F}}{m} \]
Substituting the values:
\[ \vec{a} = \frac{10 \hat{i} + 5 \hat{j}}{0.01} = 1000 \hat{i} + 500 \hat{j} \, \text{m/s}^2 \]

### Step 3: Determine the initial velocity
Since the particle starts from rest, the initial velocity \( \vec{u} = 0 \).

### Step 4: Calculate the displacement
Using the equation of motion for displacement:
\[ \vec{s} = \vec{u} t + \frac{1}{2} \vec{a} t^2 \]
Substituting the known values:
\[ \vec{s} = 0 + \frac{1}{2} (1000 \hat{i} + 500 \hat{j}) (5^2) \]
Calculating \( 5^2 = 25 \):
\[ \vec{s} = \frac{1}{2} (1000 \hat{i} + 500 \hat{j}) \times 25 \]
\[ = \frac{1}{2} (25000 \hat{i} + 12500 \hat{j}) \]
\[ = 12500 \hat{i} + 6250 \hat{j} \, \text{m} \]

### Step 5: Final result
Thus, the position of the particle at \( t = 5 \, \text{s} \) is:
\[ \vec{s} = 12500 \hat{i} + 6250 \hat{j} \, \text{m} \]