Let's analyze each formula to determine which might correctly describe the speed $v$ of a sphere sinking in a thick fluid. The speed of a sinking object in a fluid is typically determined by balancing forces, such as gravitational force and drag force, and potentially also considering buoyancy.

Key Variables:

• $C$ is a unitless constant.
• $\rho$ is the fluid's density in $\text{kg/m}^3$.
• $A$ is the sphere's cross-sectional area.
• $m$ is the mass of the sphere.
• $g$ is the gravitational field strength in $\text{N/kg}$.

Dimensional Analysis

Let’s perform dimensional analysis to check the validity of each formula.

1. $v = CA\rho g$

   • Dimensional analysis:
     $[v] = [C][A][\rho][g]$
     • $[v] = \text{m/s}$
     • $[C] = 1$ (unitless)
     • $[A] = \text{m}^2$
     • $[\rho] = \text{kg/m}^3$
     • $[g] = \text{m/s}^2$
     • Resulting dimensions: $[v] = \text{m}^2 \times \text{kg/m}^3 \times \text{m/s}^2 = \text{kg/m/s}$, which is not the correct dimension for speed. So, this is not correct.

2. $v = \frac{Cmg}{\rho A}$

   • Dimensional analysis:
     $[v] = \frac{[C][m][g]}{[\rho][A]}$
     • $[m] = \text{kg}$
     • $[g] = \text{m/s}^2$
     • $[\rho] = \text{kg/m}^3$
     • $[A] = \text{m}^2$
     • Resulting dimensions:
       $[v] = \frac{\text{kg} \times \text{m/s}^2}{\text{kg/m}^3 \times \text{m}^2} = \frac{\text{m}^4/\text{s}^2}{\text{m}} = \text{m/s}$, which is correct for speed.
       So, this formula could be correct.

3. $v = \left( \frac{Cmg}{\rho A} \right)^2$

   • This is the square of the previous formula. The dimensional analysis would yield $\text{(m/s)}^2$, which is not the correct dimension for speed. So, this is not correct.

4. $v = \left( \frac{Cmg}{\rho A} \right)^{1/2}$

   • Dimensional analysis:
     $[v] = \left( \frac{\text{kg} \times \text{m/s}^2}{\text{kg/m}^3 \times \text{m}^2} \right)^{1/2} = \left( \frac{\text{m/s}}{\text{s}} \right)^{1/2} = \text{m}^{1/2}/\text{s}^{1/2}$, which does not match the dimensions of speed. So, this is not correct.

5. None of these can be correct
   Based on the analysis, the second formula $v = \frac{Cmg}{\rho A}$ has the correct dimensions for speed. Hence, this statement is not correct.

Conclusion:

The correct formula is $v = \frac{Cmg}{\rho A}$.

Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. How does the sphere's mass affect its sinking speed?
2. What role does the fluid's density play in the sinking process?
3. How can we derive this formula using physical principles like drag and buoyancy?
4. What assumptions are made in using this formula for sinking speed?
5. How would the sinking speed change in a fluid with varying density?

Tip: When analyzing physical formulas, always check the dimensional consistency to ensure the result makes sense.