A pigeon in flight experience a force of air resistance given by `F=bv^(2)` where `v` is the flight speed and `b` is a constant. What is the maximum speed of the pigeon if its power outup is `P` ?

Options

A. `2((P)/(b))^(1//3)`
B. `((P)/(b))^(1//3)`
C. `((P)/(b))^(2//3)`
D. `2((P)/(b))^(2//3)`

Explanation

To find the maximum speed of the pigeon in flight, we start with the relationship between the power output \( P \) and the force of air resistance \( F \).

### Step 1: Understand the relationship between power, force, and velocity
The power \( P \) exerted by the pigeon must equal the power lost due to air resistance when the pigeon reaches its maximum speed. The power due to the force of air resistance is given by:
\[ P = F \cdot v \]
where \( F \) is the force of air resistance and \( v \) is the speed of the pigeon.

### Step 2: Substitute the expression for air resistance
The force of air resistance is given by:
\[ F = b v^2 \]
Substituting this into the power equation gives:
\[ P = (b v^2) \cdot v = b v^3 \]

### Step 3: Rearrange the equation to solve for \( v \)
Now, we can rearrange this equation to solve for \( v \):
\[ P = b v^3 \implies v^3 = \frac{P}{b} \]

### Step 4: Take the cube root to find maximum speed
Taking the cube root of both sides gives us the maximum speed \( v \):
\[ v = \left(\frac{P}{b}\right)^{\frac{1}{3}} \]

### Final Answer
Thus, the maximum speed of the pigeon is:
\[ v = \left(\frac{P}{b}\right)^{\frac{1}{3}} \]

---