A fighter plane is moving in a vertical circle of radius 'r'. Its minimum velocity at the highest point of the circle will be

Options

A. `sqrt( ( 1)/( 2) gr)`
B. `sqrt( 2 gr)`
C. `sqrt( gr)`
D. `sqrt( 3 gr)`

Answer

To find the minimum velocity of a fighter plane moving in a vertical circle at the highest point, we can follow these steps:

### Step 1: Understand the forces acting on the plane at the highest point
At the highest point of the vertical circle, the forces acting on the plane are:
- The gravitational force (weight) acting downwards, which is \( mg \).
- The centripetal force required to keep the plane in circular motion, which is provided by the gravitational force at this point.

### Step 2: Set up the equation for centripetal force
At the highest point, the centripetal force is given by the equation:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the plane, \( v \) is the velocity of the plane, and \( r \) is the radius of the circle.

### Step 3: Relate the gravitational force to the centripetal force
At the minimum velocity, the gravitational force provides exactly the centripetal force needed to keep the plane in circular motion. Therefore, we can set the gravitational force equal to the centripetal force:
\[ mg = \frac{mv^2}{r} \]

### Step 4: Simplify the equation
Since the mass \( m \) appears on both sides of the equation, we can cancel it out (assuming \( m \neq 0 \)):
\[ g = \frac{v^2}{r} \]

### Step 5: Solve for the velocity \( v \)
Rearranging the equation gives us:
\[ v^2 = gr \]
Taking the square root of both sides, we find:
\[ v = \sqrt{gr} \]

### Conclusion
Thus, the minimum velocity of the fighter plane at the highest point of the vertical circle is:
\[ v = \sqrt{gr} \]

### Final Answer
The minimum velocity at the highest point of the circle is \( \sqrt{gr} \).

---