Question

Force applied by water stream depends on density of water $\left( \rho \right)$ velocity of the stream $\left( v \right)$ and cross sectional area of the stream $\left( A \right)$. The expression of the force should be:
(A) $\rho Av$
(B) $\rho A{v^2}$
(C) ${\rho ^2}Av$
(D) ${\rho ^4}A = 2v$

Answer 1

Use the concept of momentum to determine the expression of force in the form of given options. Here equate the force with the momentum change rate and then substitute the formula of momentum in this equation. After this, use total mass $m = \rho Al$ for the correct answer.

Complete step by step answer:
It is given that the water density is $\rho$, the velocity of the stream is $v$ and the cross-sectional area of the stream is $A$. So we will use these notations in the calculations of force.
We know that the momentum change rate gives the value of force, so that we will use this relation.
Therefore we get,
$F = \dfrac{{dP}}{{dt}}$...... (1)
Here, $P$ is the momentum.
Write the expression of the momentum, so we get
$P = mv$
Here, $m$ is the total mass of the water and $v$ is the stream's velocity.
Substitute $P = = mv$ in the equation (1).
$F = \dfrac{{d\left( {mv} \right)}}{{dt}}$...... (2)
We know that the expression of the water's total mass is $m = \rho Al$, here $l$ the stream's length. So, we will use this expression in equation (2) for the calculation of force.
Therefore, we get
$\begin{array}{l} F = \dfrac{{d\left( {\rho Al} \right)v}}{{dt}}\\\ F = \dfrac{{dl}}{{dt}}\rho Av \end{array}$
The change in the stream's length with time gives information about the stream's velocity, so we will use $dl/dt = v$ in the above equation. So, the above equation of force becomes
$\begin{array}{l} F = v\rho Av\\\ F = \rho A{v^2} \end{array}$
Therefore, the force's expression should be $F = \rho A{v^2}$, and option (B) is correct.

Note: Remember the expressions of momentum and total mass for the correct calculation of the force. Also, use the correct notations of the various terms in the calculation. If we put an incorrect notation in the calculation, our answer may vary from the given options.