Physics 9702/21 — May/June 2019

Background Concept

The given relationship is

$$v = \sqrt{\frac{kP}{\rho}}$$

Squaring both sides removes the square root:

$$v^2 = \frac{kP}{\rho}$$

This is then rearranged to make $k$ the subject:

$$k = \frac{v^2\rho}{P}$$

Here $k$ is stated to be dimensionless (no units), which is consistent because $\frac{P}{\rho}$ has units of $\text{m}^2\text{s}^{-2}$ and its square root has units $\text{m s}^{-1}$.

Understanding the Question

You are given measured values of:

• $v = 3.3 \times 10^2\ \text{m s}^{-1}$
• $P = 9.9 \times 10^4\ \text{Pa}$
• $\rho = 1.29\ \text{kg m}^{-3}$

and asked to calculate the constant $k$ using the equation.

Approach

1. Rearrange the equation to get $k$ in terms of $v$, $P$, and $\rho$.
2. Substitute the numerical values.
3. Calculate carefully, keeping track of powers of ten.
4. Round sensibly (typically to 2 s.f. from the given data).

Step-by-Step Reasoning

Start with

$$v = \sqrt{\frac{kP}{\rho}}$$

Square both sides:

$$v^2 = \frac{kP}{\rho}$$

Rearrange for $k$ by multiplying by $\rho$ and dividing by $P$:

$$k = \frac{v^2\rho}{P}$$

Now substitute values:

1. Square $v$:

$$(3.3 \times 10^2)^2 = 3.3^2 \times 10^4 = 10.89 \times 10^4 = 1.089 \times 10^5$$

1. Multiply by $\rho$:

$$1.089 \times 10^5 \times 1.29 = 1.40481 \times 10^5$$

1. Divide by $P$:

$$\frac{1.40481 \times 10^5}{9.9 \times 10^4} = \left(\frac{1.40481}{9.9}\right) \times 10^{1} \approx 1.42$$

So $k \approx 1.42$, which rounds to $1.4$ (2 s.f.).

Key Takeaways

• To remove a square root, square both sides before rearranging.
• When squaring standard form, square the number and double the power of ten.
• Check that the final value is reasonable and matches the expected unit statement (here: no units).

Common Mistakes

• Forgetting to square $v$.
• Rearranging incorrectly (e.g. using $k = \frac{v^2P}{\rho}$).
• Power-of-ten errors when squaring $3.3 \times 10^2$.

Things to Be Careful About

• Significant figures: inputs are mostly 2 s.f., so quoting $k$ to 2 s.f. is appropriate.
• Don’t attach units to $k$ because the question states it has none.

Background Concept

When quantities are multiplied or divided, percentage (fractional) uncertainties add.

If

$$Q = \frac{AB}{C}$$

then

$$\frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C}$$

Also, if a quantity is raised to a power, the percentage uncertainty is multiplied by that power. For

$$Q = A^n$$

then

$$\frac{\Delta Q}{Q} = |n|\frac{\Delta A}{A}$$

Finally, to convert from percentage uncertainty to absolute uncertainty:

$$\Delta Q = Q \times \left(\frac{\text{percentage}}{100}\right)$$

Understanding the Question

You already found $k$ in part (i). Now you must use the percentage uncertainties in $v$, $P$, and $\rho$ to find:

• the absolute uncertainty in $k$, and
• quote $k \pm \Delta k$ to a sensible number of significant figures.

Given uncertainties:

• $v$: $\pm 3\%$
• $P$: $\pm 2\%$
• $\rho$: $\pm 4\%$

Approach

1. Write $k$ in terms of the measured variables: $k = \frac{v^2\rho}{P}$.
2. Add percentage uncertainties, remembering the factor of 2 because of $v^2$.
3. Convert the final percentage uncertainty into an absolute uncertainty using your value of $k$.
4. Round the uncertainty (usually to 1 s.f.), then round $k$ to the same decimal place.

Step-by-Step Reasoning

From the rearranged equation:

$$k = \frac{v^2\rho}{P}$$

1) Percentage uncertainty in $v^2$

Since $v$ has $\pm 3\%$ uncertainty and $v$ is squared,

$$\%\Delta (v^2) = 2 \times 3\% = 6\%$$

2) Combine uncertainties for multiplication/division

$k$ is proportional to $v^2$ and $\rho$ and inversely proportional to $P$, so we add all their percentage uncertainties:

$$\%\Delta k = 6\% + 4\% + 2\% = 12\%$$

3) Convert to absolute uncertainty

Using $k \approx 1.42$ from part (i):

$$\Delta k = 0.12 \times 1.42 = 0.1704$$

4) Quote to appropriate significant figures

Uncertainty $0.1704$ rounds to $0.2$ (1 s.f.). Then quote $k$ to the same decimal place (tenths):

$$k = 1.4 \pm 0.2$$

Key Takeaways

• For products/quotients: add percentage uncertainties.
• For powers: multiply percentage uncertainty by the power.
• Round uncertainty to 1 s.f. (or 2 s.f. if it begins with 1 or 2 is sometimes accepted), then round the value to match.

Common Mistakes

• Forgetting the factor of 2 for $v^2$.
• Subtracting the percentage uncertainty for the denominator (you still add it).
• Quoting too many significant figures in the uncertainty (e.g. $\pm 0.1704$) or not matching decimal places (e.g. $1.42 \pm 0.2$).

Things to Be Careful About

• Use the value of $k$ from (i) (or your own calculated value) when finding the absolute uncertainty.
• Final presentation: value and absolute uncertainty should have consistent decimal places.
• Ensure you are using percentage as a fraction correctly: $12\% = 0.12$.