A cat runs along a straight line with constant velocity of magnitude v. A dog chases the cat such that the velocity of dog is always directed towards the cat. The speed of dog is `u` and always constant .At the instant both are separated by distance x and their velocities are mutually perpendicular, the magnitude of acceleration of dog is.

Options

A. `(uv)/(x)`
B. `(u^2)/(x)`
C. `(v^2)/(x)`
D. `(u^2+v^2)/(x)`

Explanation

To solve the problem, we need to analyze the motion of the cat and the dog. The cat is moving with a constant velocity \( v \), and the dog is chasing the cat with a constant speed \( u \), always directed towards the cat. At the instant when they are separated by a distance \( x \) and their velocities are mutually perpendicular, we need to find the magnitude of the dog's acceleration.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the cat move along the x-axis with a constant velocity \( v \).
- The dog is at a distance \( x \) from the cat and is moving towards the cat with a constant speed \( u \).
- At the instant considered, the velocity of the dog is perpendicular to the velocity of the cat.

2. **Identifying Velocities**:
- The velocity of the cat can be represented as \( \vec{v}_{\text{cat}} = v \hat{i} \).
- The velocity of the dog can be represented as \( \vec{v}_{\text{dog}} = u \hat{j} \) (where \( \hat{j} \) is perpendicular to \( \hat{i} \)).

3. **Using Geometry**:
- The dog moves towards the cat, creating a right triangle where the hypotenuse is the line of sight from the dog to the cat, and the legs are the components of the velocities.
- The distance \( x \) remains constant while the dog is approaching the cat.

4. **Finding the Angle**:
- Let \( \theta \) be the angle between the line connecting the dog and cat and the direction of the dog's velocity.
- The relationship between the velocities can be expressed using trigonometry:
\[ \sin(\theta) = \frac{v}{u} \]

5. **Calculating the Acceleration**:
- The acceleration of the dog can be derived from the change in the direction of the velocity vector as it continuously adjusts to chase the cat.
- The acceleration \( a \) of the dog can be expressed as:
\[ a = \frac{u v}{x} \]
- This is derived from the fact that the component of the dog's velocity directed towards the cat changes as the dog moves closer, and the relationship between the velocities and distance.

6. **Final Result**:
- Therefore, the magnitude of the acceleration of the dog is:
\[ a = \frac{u v}{x} \]