When at t = 0, a particle at (1, 0, 0) moves towards (4, 4, 12) with a constant speed of 65 m/s. The position of the particle is measured in meters and time in sec. Assuming constant velocity, the x
co-ordinate of the particle at t = 2 sec is:

Explanation

To find the x-coordinate of the particle at \( t = 2 \) seconds, we can follow these steps:

### Step 1: Determine the Initial and Final Positions
The particle starts at the position \( (1, 0, 0) \) and moves towards the position \( (4, 4, 12) \).

### Step 2: Calculate the Displacement Vector
The displacement vector \( \vec{d} \) from the initial position to the final position can be calculated as follows:
\[ \vec{d} = (4 - 1) \hat{i} + (4 - 0) \hat{j} + (12 - 0) \hat{k} = 3 \hat{i} + 4 \hat{j} + 12 \hat{k} \]

### Step 3: Calculate the Magnitude of the Displacement
The magnitude of the displacement vector \( |\vec{d}| \) is given by:
\[ |\vec{d}| = \sqrt{(3^2) + (4^2) + (12^2)} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \]

### Step 4: Find the Unit Vector in the Direction of Motion
The unit vector \( \hat{u} \) in the direction of the displacement is:
\[ \hat{u} = \frac{\vec{d}}{|\vec{d}|} = \frac{3 \hat{i} + 4 \hat{j} + 12 \hat{k}}{13} \]

### Step 5: Calculate the Velocity Vector
The velocity vector \( \vec{v} \) of the particle, moving at a constant speed of \( 65 \) m/s, is given by:
\[ \vec{v} = 65 \hat{u} = 65 \left( \frac{3 \hat{i} + 4 \hat{j} + 12 \hat{k}}{13} \right) = \frac{195}{13} \hat{i} + \frac{260}{13} \hat{j} + \frac{780}{13} \hat{k} = 15 \hat{i} + 20 \hat{j} + 60 \hat{k} \]

### Step 6: Calculate the Position Vector at \( t = 2 \) seconds
The position vector \( \vec{r}(t) \) at time \( t \) can be calculated using:
\[ \vec{r}(t) = \vec{r}_0 + \vec{v} t \]
Where \( \vec{r}_0 = (1, 0, 0) \) is the initial position. Thus, at \( t = 2 \) seconds:
\[ \vec{r}(2) = (1, 0, 0) + (15 \hat{i} + 20 \hat{j} + 60 \hat{k}) \cdot 2 = (1 + 30) \hat{i} + (0 + 40) \hat{j} + (0 + 120) \hat{k} = 31 \hat{i} + 40 \hat{j} + 120 \hat{k} \]

### Step 7: Extract the x-coordinate
The x-coordinate of the particle at \( t = 2 \) seconds is:
\[ x = 31 \]

Thus, the x-coordinate of the particle at \( t = 2 \) seconds is **31 meters**.

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