Question

A pigeon in flight experiences a force of air resistance given by $F = b{v^2}$ where v is the flight speed and b is a constant. What is the maximum speed of the pigeon if its power output is P?
A. $2{\left( {\dfrac{P}{b}} \right)^{\dfrac{1}{3}}}$
B. ${\left( {\dfrac{P}{b}} \right)^{\dfrac{1}{3}}}$
C. ${\left( {\dfrac{P}{b}} \right)^{\dfrac{2}{3}}}$
D. $2{\left( {\dfrac{P}{b}} \right)^{\dfrac{2}{3}}}$

Answer 1

The maximum speed of the pigeon will be obtained when the power spent by air resistance is equal to the power output of the pigeon. Equate the power of air resistance with the power output of the pigeon. Power is the product of Force and speed.

Complete step by step answer:
We are given that a pigeon in flight experiences a force of air resistance given by $F = b{v^2}$ where v is the speed of the flight and b is a constant.
We have to calculate the maximum speed of the pigeon if its power output is P.
To obtain the maximum speed by the pigeon, ${P_{pigeon}}$ (power output of pigeon) must be equal to ${P_{air\\_resistance}}$ (power spent by air resistance)
We are already given that the power output of the Pigeon is P.
We have to calculate power spent by air resistance.
Power spent by air resistance is ${P_{air\\_resistance}} = F \times v$
We know that the value of $F = b{v^2}$
Therefore,
${P_{air\\_resistance}} = b{v^2} \times v = b{v^3} \to eq\left( 1 \right)$
On equating equation 1 with the power output of pigeon, we get
$P = b{v^3} \\\ \to {v^3} = \dfrac{P}{b} \\\$
On sending the cube power from left hand side to right hand side, we get
$\therefore v = {\left( {\dfrac{P}{b}} \right)^{\dfrac{1}{3}}}$
Therefore, the maximum speed of the pigeon is ${\left( {\dfrac{P}{b}} \right)^{\dfrac{1}{3}}}$

Hence, the correct option is Option B.

Note: Power is also defined as the rate of doing work or the rate of utilizing energy, $P = \dfrac{W}{t},P = \dfrac{E}{t}$. Energy (work) is the product of force and displacement. So power will be $P = \dfrac{{F \times d}}{t}$. Where displacement over time is known as velocity, therefore the power will be $P = F \times v$. In the question, we are given with the force and velocity so we have used the obtained formula, when the energy or the work done is given then we have to use the first formula to solve for power.