Figure 7 shows apparatus used to investigate the rate in which water flows through a hollowed cylindrical tube of internal diameter $d$ and length $L$ - AQA - A-Level Physics - Question 2 - 2019 - Paper 3

Two suitable instruments for this experiment could be a:

1. Graduated cylinder to accurately measure the volume of water collected.
2. Stopwatch to time the duration for which the water flows into the graduated cylinder.

To reduce uncertainty in the measurements, the student should:

1. Take multiple measurements for the same experiment and calculate an average value to minimize the random errors.
2. Use a more precise measuring device, such as a burette, to measure the volume of water accurately.
3. Ensure that all apparatus is level to prevent any discrepancies in water flow due to gravitational effects.

The unit for $k$ can be derived from the equation $Q = k \sqrt{\frac{2gh}{\eta}}$. Rearranging gives:

$k = Q \sqrt{\frac{\eta}{2gh}}$

Here, $Q$ has units of $m^3/s$, $h$ is in meters, $g$ is in $m/s^2$, and $\eta$ has units of $Pa.s = N.s.m^{-2}$. Therefore, the unit for $k$ is $N m^{-3} s^{-1}$.

The percentage uncertainty can be calculated using the formula:

$\text{Percentage Uncertainty} = \left(\frac{\Delta Q}{Q}\right) \times 100\% + \left(\frac{\Delta T}{T}\right) \times 100\%$

Substituting the values:

Assuming $Q = 20$L (which is 0.02 m³) and $T = 4.5 \times 10^{-3}$ s, we get:

$\text{Percentage Uncertainty} = \left(\frac{1.2}{0.02}\right) \times 100 + \left(\frac{1.5}{4.5 \times 10^{-3}}\right) \times 100\% = 19.5\%$

The student could not use the glass tube viewed vertically because the water would not rise correctly as the angle of observation could lead to parallax errors. Viewing the tube from an angle can distort the perception of the water level, leading to inaccurate distance measurements.

To show that $T = T_0 e^{-kt}$, we start with the differential equation relating to exponential decay:

$\frac{dT}{dt} = -kT$

Solving this differential equation gives the solution form:

$T(t) = T_0 e^{-kt}$

With $T_0$ as the initial value of T when $t=0$.

To demonstrate this graphically, the student should create a line on Figure 13 representing the new data points corresponding to the recorded values $y_i$. The graph should show a decrease in $y$ over increasing water levels, similar to an exponential decay curve in Figure 10, but adjusted to fit the measurements recorded when the tube is inclined at 30 degrees.