A stone is tied to a string and swings with uniform motion in a horizontal circle. The string breaks and at a time t, the stone is displaced `Deltar=3 hat(i)+4hat(j)-5hat(k)` metres. (The positive z-axis is vertically up) Select the correct alternative.

Options

A. the time t is 1 s
B. the time t is 0.5 s
C. the speed of the stone while in circular motion is 5 m/s
D. the speed of the stone while in circular motion is `5 sqrt(2)m//s`

Answer

To solve the problem, we need to analyze the motion of the stone after the string breaks. The stone is initially moving in a horizontal circle, and when the string breaks, it will move in a straight line due to inertia. We will break down the displacement vector and analyze the vertical and horizontal components.

### Step-by-Step Solution:

1. **Identify the Displacement Vector**:
The displacement vector is given as:
\[ \Delta \mathbf{r} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k} \text{ meters} \]
Here, \(3 \hat{i}\) represents the displacement in the x-direction, \(4 \hat{j}\) represents the displacement in the y-direction, and \(-5 \hat{k}\) indicates a downward displacement in the z-direction.

2. **Vertical Displacement**:
The vertical component of the displacement is \(-5 \hat{k}\), which means the stone has moved 5 meters downward. Since the positive z-axis is vertically up, this indicates that the stone has fallen 5 meters.

3. **Calculate Time of Fall**:
Using the equation of motion for vertical displacement:
\[ h = \frac{1}{2} g t^2 \]
where \(h = 5\) meters (downward), and \(g = 10 \, \text{m/s}^2\) (acceleration due to gravity), we can solve for \(t\):
\[ 5 = \frac{1}{2} \cdot 10 \cdot t^2 \]
\[ 5 = 5 t^2 \]
\[ t^2 = 1 \implies t = 1 \text{ second} \]

4. **Horizontal Displacement**:
The horizontal displacement in the xy-plane is given by the components \(3 \hat{i}\) and \(4 \hat{j}\). The speed of the stone in the horizontal motion can be calculated using the Pythagorean theorem:
\[ v = \sqrt{(3^2 + 4^2)} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

5. **Conclusion**:
From the calculations, we find:
- The time taken to fall 5 meters is \(1\) second.
- The speed of the stone while in circular motion before the string broke is \(5 \, \text{m/s}\).