`q=1/2nv^(2)`
The dynamic pressure q generated by a fluid moving with velocity v can be found using the formula above, where n is the constant density of the fluid. An aeronautical engineer uses the formula to find the dynamic pressure of a fluid moving with velocity v and the same fluid moving with velocity 1.5v. What is the ratio of the dynamic pressure of the faster fluid to the dynamic pressure of the slower fluid?

Explanation

To solve the problem, we need to find the ratio of the dynamic pressures of two fluids moving at different velocities using the given formula for dynamic pressure:

\[ q = \frac{1}{2} n v^2 \]

where:
- \( q \) is the dynamic pressure,
- \( n \) is the constant density of the fluid,
- \( v \) is the velocity of the fluid.

### Step 1: Calculate the dynamic pressure for the faster fluid

The faster fluid is moving with a velocity of \( 1.5v \). We can substitute this value into the dynamic pressure formula:

\[ q_1 = \frac{1}{2} n (1.5v)^2 \]

Calculating \( (1.5v)^2 \):

\[ (1.5v)^2 = 2.25v^2 \]

Now substituting back into the equation for \( q_1 \):

\[ q_1 = \frac{1}{2} n (2.25v^2) = \frac{2.25}{2} n v^2 = 1.125 n v^2 \]

### Step 2: Calculate the dynamic pressure for the slower fluid

The slower fluid is moving with a velocity of \( v \). We substitute this value into the dynamic pressure formula:

\[ q_2 = \frac{1}{2} n v^2 \]

### Step 3: Find the ratio of the dynamic pressures

Now we need to find the ratio of the dynamic pressure of the faster fluid \( q_1 \) to the dynamic pressure of the slower fluid \( q_2 \):

\[ \text{Ratio} = \frac{q_1}{q_2} = \frac{1.125 n v^2}{\frac{1}{2} n v^2} \]

### Step 4: Simplify the ratio

The \( n \) and \( v^2 \) terms cancel out:

\[ \text{Ratio} = \frac{1.125}{\frac{1}{2}} = 1.125 \times 2 = 2.25 \]

### Step 5: Convert to a fraction

To express \( 2.25 \) as a fraction:

\[ 2.25 = \frac{9}{4} \]

### Conclusion

Thus, the ratio of the dynamic pressure of the faster fluid to the dynamic pressure of the slower fluid is:

\[ \text{Ratio} = 9:4 \]