# Investigating Terminal Velocity ## Experimental Overview - The terminal velocity of a sphere falling through a viscous liquid can be measured using laboratory equipment - The objective is to determine the steady-state speed achieved when drag forces balance the weight of the falling object - This practical demonstrates the relationship between forces, motion and fluid resistance ## Variables **Independent Variable:** - Vertical distance between measurement markers (measured in metres) **Dependent Variable:** - Time taken for the sphere to pass between consecutive markers (measured in seconds) **Control Variables:** - Same viscous liquid throughout (e.g. glycerol, wallpaper paste, or golden syrup) - Same sphere used for all measurements (consistent mass, diameter, and surface properties) - Same experimental temperature (viscosity changes with temperature)

Using a highly viscous fluid slows down the sphere's motion, making timing measurements significantly more accurate.

## Equipment and Purpose | **Equipment** | **Purpose** | |---------------|-------------| | Transparent cylinder (tall measuring cylinder) | Contains the viscous liquid; height determines how long the sphere travels at terminal velocity | | Viscous liquid (glycerol, thick oil, or wallpaper paste) | Provides resistance that allows terminal velocity to be reached within a reasonable distance | | Steel or glass spheres | Test object that falls through the fluid; density must exceed that of the liquid | | Rubber bands or permanent marker | Mark reference points at known vertical intervals for timing | | Metre ruler | Measure precise distances between reference markers | | Stopwatch | Record time intervals as sphere passes each marker | | Retort stand with clamp | Secure the cylinder vertically to prevent movement during measurements | | Strong magnet | Retrieve sphere from bottom of cylinder without contamination | **Resolution of Measuring Instruments:** - Metre ruler: 1 mm (0.001 m) - Digital stopwatch: 0.01 s ## Experimental Procedure 1. **Setup the apparatus** - Secure the transparent cylinder vertically using the retort stand and clamp - Fill the cylinder with viscous liquid, leaving approximately 5 cm space at the top - Allow the liquid to settle for several minutes to eliminate air bubbles 2. **Mark measurement intervals** - Use a metre ruler to measure equal vertical intervals down the cylinder (e.g. every 10 cm) - Wrap elastic bands tightly around the cylinder at these positions, or draw clear marks with permanent marker - Record the precise distance between each consecutive pair of markers - Ensure the first marker is sufficiently far from the liquid surface (at least 15-20 cm) to allow the sphere to accelerate to terminal velocity 3. **Release the sphere and collect data** - Hold the sphere just above the liquid surface, centred over the cylinder - Release the sphere from rest (no downward push) - Start the stopwatch when the sphere passes the first marker - Record the time at each subsequent marker using the lap function on the stopwatch - Observe that the sphere eventually falls at constant speed (visual confirmation of terminal velocity) 4. **Retrieve and repeat** - Use the magnet to extract the sphere from the bottom of the cylinder - Wipe the sphere carefully to remove excess liquid - Repeat the entire procedure at least four times to obtain reliable mean values 5. **Calculate time intervals** - Determine the time taken to travel between each consecutive pair of markers - Calculate the mean time for each interval from your repeated measurements

Terminal velocity is reached when drag force equals weight, resulting in zero net force and constant velocity.

## Data Analysis **Calculating Velocities:** - For each pair of markers, calculate the average speed using: $$v = \frac{\Delta s}{\Delta t}$$ where $\Delta s$ is the distance between markers and $\Delta t$ is the average time interval **Example Table Structure:** | Distance Between Markers (m) | Time (s) - Trial 1 | Time (s) - Trial 2 | Time (s) - Trial 3 | Time (s) - Trial 4 | Mean Time (s) | Speed (m s$^{-1}$) | |------------------------------|---------------------|---------------------|---------------------|---------------------|---------------|-------------------| | 0.00 - 0.10 | | | | | | | | 0.10 - 0.20 | | | | | | | | 0.20 - 0.30 | | | | | | | | 0.30 - 0.40 | | | | | | | **Graphical Analysis:** - Plot a graph of velocity (y-axis) against time (x-axis) - Draw a smooth curve through the data points - The graph should show velocity increasing rapidly at first, then levelling off - The terminal velocity is the constant velocity where the graph becomes horizontal (zero gradient) - Read the terminal velocity directly from the plateau region of the graph The typical shape of the velocity-time graph is shown below, illustrating how velocity increases rapidly initially before asymptotically approaching terminal velocity:

When the velocity between consecutive markers becomes constant (within experimental uncertainty), the sphere has reached terminal velocity.

## Improving Accuracy and Precision **Systematic Improvements:** - **Use a taller cylinder** - Provides greater distance for the sphere to travel at terminal velocity - More data points can be collected in the constant velocity region - Reduces the proportional effect of the acceleration phase on results - **Select a more viscous liquid** - Slower motion makes visual tracking easier - Timing measurements have lower percentage uncertainty - Terminal velocity is reached over a shorter distance - **Increase spacing between markers** - Larger distance intervals reduce percentage uncertainty in distance measurements - Longer time intervals reduce percentage uncertainty in timing - However, fewer data points may result, so balance is needed - **Use light gates and data logging** - Eliminates human reaction time errors - Records exact times electronically - Can measure very short time intervals accurately **Random Error Reduction:** - **Conduct multiple trials (minimum of 4)** - Calculate mean values to reduce the impact of anomalous results - Identify and investigate any outliers - Allows calculation of uncertainty in measurements - **Alternative timing method - ticker tape** - Attach ticker tape to the sphere (if practical) - Ticker timer creates dots at regular time intervals (e.g. 0.02 s) - Measure distances between dots to calculate velocities - Eliminates stopwatch reaction time uncertainty - Provides continuous velocity data throughout the fall - **Video analysis technique** - Record the falling sphere with a high-frame-rate camera - Use video analysis software to track position frame-by-frame - Calculate velocities from position-time data - Allows detailed analysis of the acceleration phase

Ensure the sphere is always released from rest at the same position to maintain consistency between trials.

## Sources of Uncertainty **Measurement Uncertainties:** - **Distance measurements:** ±1 mm uncertainty from ruler resolution - **Time measurements:** ±0.01 s from stopwatch resolution, plus human reaction time (±0.2 s) - **Sphere positioning:** May not fall perfectly centrally, causing slight deviations in path **Procedural Challenges:** - Difficulty in determining the exact moment the sphere crosses each marker (parallax error) - Liquid may flow or create currents that affect sphere motion - Temperature variations during the experiment change liquid viscosity - Sphere may rotate or tumble slightly, affecting drag forces **Minimising Uncertainties:** - View markers at eye level to reduce parallax errors - Allow sufficient time between trials for liquid to settle - Conduct experiment in a temperature-controlled environment - Use larger distance intervals to reduce percentage uncertainty - Repeat measurements multiple times and calculate means