A ladder placed on a smooth floor slips. If at a given instant the velocity with which the ladder is slipping on the floor is `v_1`, and the velocity of that part of ladder which is touching the wall is `v_1`, then the velocity of the centre of the ladder at that instant is:

Options

A. `v_1`
B. `v_2`
C. `(v_1+v_2)/2`
D. `sqrt(v_1^2+v_2^2)/2`

Explanation

To solve the problem of finding the velocity of the center of the ladder at the instant it is slipping, we can follow these steps:

### Step 1: Understand the Setup
We have a ladder leaning against a wall and resting on a smooth floor. As the ladder slips, the point touching the floor moves with a velocity \( v_1 \) horizontally, and the point touching the wall moves with a velocity \( v_2 \) vertically downward.

### Step 2: Define the Coordinates
Let's define the coordinates of the points:
- The point on the floor (bottom of the ladder) is at coordinates \( (x_0, 0) \).
- The point against the wall (top of the ladder) is at coordinates \( (0, y_0) \).
- The center of the ladder will be at coordinates \( \left(\frac{x_0}{2}, \frac{y_0}{2}\right) \).

### Step 3: Relate the Velocities
The velocity of the bottom point (slipping on the floor) is given as \( v_1 \), and the velocity of the top point (slipping against the wall) is given as \( v_2 \).
- The rate of change of \( x_0 \) (horizontal position) is \( \frac{dx_0}{dt} = v_1 \).
- The rate of change of \( y_0 \) (vertical position) is \( \frac{dy_0}{dt} = -v_2 \) (negative because it is moving downward).

### Step 4: Calculate the Velocity of the Center of the Ladder
The velocity of the center of the ladder can be calculated by taking the average of the velocities of the two points:
- The velocity in the x-direction (horizontal) for the center is \( \frac{1}{2} \frac{dx_0}{dt} = \frac{v_1}{2} \).
- The velocity in the y-direction (vertical) for the center is \( \frac{1}{2} \frac{dy_0}{dt} = \frac{-v_2}{2} \).

Thus, the velocity of the center of the ladder can be expressed as:
\[ \vec{V}_{\text{center}} = \frac{v_1}{2} \hat{i} - \frac{v_2}{2} \hat{j} \]

### Step 5: Calculate the Magnitude of the Velocity
To find the magnitude of the velocity of the center of the ladder, we use the Pythagorean theorem:
\[ |\vec{V}_{\text{center}}| = \sqrt{\left(\frac{v_1}{2}\right)^2 + \left(-\frac{v_2}{2}\right)^2} \]
\[ |\vec{V}_{\text{center}}| = \sqrt{\frac{v_1^2}{4} + \frac{v_2^2}{4}} = \frac{1}{2} \sqrt{v_1^2 + v_2^2} \]

### Final Answer
The velocity of the center of the ladder at that instant is:
\[ \frac{1}{2} \sqrt{v_1^2 + v_2^2} \]