The cylindrical tube of a spray pump has radius $R$, one end of which has $n$ fine holes, each of radius $r$.

Question

The cylindrical tube of a spray pump has radius $R$, one end of which has $n$ fine holes, each of radius $r$. If the speed of the liquid in the tube is $v$, then the speed of ejection of the liquid through the holes will be:

Accepted Answer

1. **Identify the Principle:** The problem is based on the **Equation of Continuity** for an incompressible fluid, which states that the rate of flow (volume flux) remains constant throughout the pipe ($A_1 v_1 = A_2 v_2$) .
2. **Determine Inlet Conditions:**
   - Radius of the main tube = $R$
   - Area of cross-section ($A_1$) = $\pi R^2$
   - Speed of liquid ($v_1$) = $v$
   - Volume flow rate entering = $A_1 v_1 = \pi R^2 v$
3. **Determine Outlet Conditions:**
   - Number of holes = $n$
   - Radius of each hole = $r$
   - Total Area of cross-section ($A_2$) = $n 	imes \pi r^2$
   - Speed of ejection ($v_2$) = ?
   - Volume flow rate exiting = $A_2 v_2 = n \pi r^2 v_2$
4. **Apply Conservation of Mass:**
   Equating flow rates:
   $$ \pi R^2 v = n \pi r^2 v_2 $$
   $$ v_2 = \frac{\pi R^2 v}{n \pi r^2} $$
   $$ v_2 = \frac{v R^2}{n r^2} $$

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