Practical Investigation: Validating Projectile Motion

What we expect to observe based on the model

Practical setup: ball bearing rolls from ramp exit at height $h$ and lands at horizontal range $R$.

Detailed equipment layout showing retort stand, ramp exit, measuring positions and landing zone.

Aim: To determine the relationship between the launch height and the horizontal range of a projectile launched horizontally, and to compare the experimental results with the theoretical model.

Assumptions of the model:

• The vertical acceleration is constant and equal to $g = 9.8\ \text{m/s}^2$ downward
• Air resistance is negligible
• The launch velocity is horizontal (launch angle $\theta = 0°$)

For a horizontal launch from height $h$ with speed $v_x$:

If $v_x$ and $g$ are constant, then $R \propto \sqrt{h}$. A graph of $R$ versus $\sqrt{h}$ should be a straight line through the origin with gradient $v_x\sqrt{2/g}$.

For a horizontal launch from height $h$: time of flight $t = \sqrt{2h/g}$ and range $R = v_x\sqrt{2h/g}$, so $R \propto \sqrt{h}$. Plot $R$ vs $\sqrt{h}$, a straight line through the origin with gradient $v_x\sqrt{2/g}$ validates the model.

Pause, copy the highlighted theoretical prediction and the graph strategy into your book before moving on.

A reproducible procedure for collecting valid data

We just saw the theoretical prediction: $R \propto \sqrt{h}$. That raises a question: how do we design the experiment to test this fairly and reproducibly? This card answers it → variables, equipment, and a step-by-step procedure that controls $v_x$ at each height.

Equipment:

• Ball bearing or small solid sphere (minimise air resistance)
• Smooth ramp or track with horizontal exit section
• Retort stand and clamp
• Metre ruler (±1 mm)
• Carbon paper and white paper (or motion sensor / video camera)
• Balance (to measure mass, optional)
• Stopwatch (±0.01 s, optional for time-of-flight check)

Variables:

• Independent: Launch height $h$ (measured from floor to exit point)
• Dependent: Horizontal range $R$ (measured from base of ramp to landing point)
• Controlled: Launch speed $v_x$ (same release position on ramp), same ball, same surface, minimal air movement

Procedure:

1. Set up the ramp so the exit is perfectly horizontal. Check with a spirit level.
2. Place carbon paper over white paper on the floor to mark landing positions.
3. Release the ball from a fixed position on the ramp. Do not push, let it roll from rest to ensure consistent speed.
4. Measure the vertical height $h$ from the floor to the centre of the ball at the exit point.
5. Measure the horizontal range $R$ from the point directly below the exit to the centre of the landing mark.
6. Repeat for at least 5 different heights, keeping the release position constant.
7. At each height, perform at least 3 trials and average the range.

IV = launch height $h$ (m); DV = horizontal range $R$ (m); controlled = same release point on ramp (constant $v_x$), same ball. Use minimum 5 heights, 3 trials each; carbon paper marks the first-contact point.

Add the highlighted variables table to your notes before the check below.

Record your primary data here

We just saw the method, 5 heights, 3 trials each, carbon paper landing marks. That raises a question: how should the results be organised to make analysis straightforward? This card answers it → a table with $\sqrt{h}$ pre-computed and uncertainty as half-spread of the three trials.

Worked sample data (for processing practice only; this does not replace primary data):

Record raw distances to the precision justified by the measuring device. For an analogue scale with smallest division $d$, a typical reading uncertainty is $\pm d/2$; repeat half-spread describes trial scatter and is a different quantity.

Results table must include: $h$, $\sqrt{h}$, three range trials, mean $\bar{R}$, and uncertainty $\Delta R = (R_\text{max} - R_\text{min})/2$. The $\sqrt{h}$ column is essential, it linearises the relationship for graphing.

Pause, draw the results table with these column headings in your book before the next section.

Compare experiment to theory

We just saw how to set up the results table with $\sqrt{h}$ values. That raises a question: how do we extract a launch speed from the graph and check how well the model fits? This card answers it → measure the gradient of the $R$ vs $\sqrt{h}$ line and compare it to $m_\text{theory} = v_x\sqrt{2/g}$.

Step 1, Plot the graph

Plot $\bar{R}$ (vertical axis) versus $\sqrt{h}$ (horizontal axis), with vertical uncertainty bars. Draw a line of best fit; do not force it through the origin unless the evidence and uncertainty justify that choice.

Step 2, Determine the gradient

The theoretical gradient is:

$$m_{\text{theory}} = v_x \sqrt{\dfrac{2}{g}}$$

From your graph, calculate the experimental gradient $m_{\text{exp}}$ using:

$$m_{\text{exp}} = \dfrac{\Delta R}{\Delta \sqrt{h}}$$

Step 3, Compare

Calculate the percentage difference:

$$\%\ \text{difference} = \dfrac{|m_{\text{exp}} - m_{\text{theory}}|}{m_{\text{theory}}} \times 100\%$$

Step 4, Calculate launch speed from data

Rearranging the gradient formula:

$$v_x = m_{\text{exp}} \sqrt{\dfrac{g}{2}}$$

Compare this calculated $v_x$ to any independent measurement of launch speed (e.g., from a motion sensor or timing gate).

Plot $\bar{R}$ vs $\sqrt{h}$ with uncertainty bars. The gradient $m=\Delta R/\Delta\sqrt{h}=v_x\sqrt{2/g}$ has units $\text{m}^{1/2}$ and gives $v_x=m\sqrt{g/2}$. Agreement supports the model only within the tested conditions and uncertainty.

Add the highlighted analysis steps and acceptance criterion to your notes before the check below.

Working Scientifically, identify and address limitations

We just saw how to validate the model by comparing experimental and theoretical gradients. That raises a question: what causes discrepancies, and how can the experiment be improved? This card answers it → classify errors as systematic or random, then propose targeted improvements.

Systematic errors:

• Ramp not perfectly horizontal: A slight upward angle increases range; downward decreases it. Check with a spirit level.
• Air resistance: Significant for light objects or high speeds. Use a dense ball bearing.
• Release position inconsistent: If the ball is pushed or released from slightly different points, $v_x$ varies. Use a release gate or marked release line.

Random uncertainty:

• Landing spread: Small variations in release and first-contact position produce trial-to-trial scatter. Carbon paper helps mark first contact.
• Timing uncertainty: If measuring time of flight with a stopwatch, human reaction time (~0.2 s) is a significant source of error.

Model discrepancy: Air resistance and ball rotation are physical effects omitted by the ideal horizontal-launch model. A repeatable departure caused by an omitted effect is not merely measurement error; it identifies a limitation of the model.

Reliability:

• Repeating trials at each height and averaging reduces random error.
• A spread of at least 5 different heights tests the model across a range of conditions.

Improvements:

• Use a video camera or motion sensor to measure $v_x$ independently and compare.
• Conduct the experiment in a vacuum chamber (advanced) to eliminate air resistance.
• Use a photogate at the exit to measure $v_x$ directly and verify the constant-speed assumption.

Random uncertainty produces scatter and is estimated with repeats; systematic bias shifts results consistently and is not removed by averaging; model discrepancy comes from omitted physics such as drag or rotation. A photogate independently measures $v_x$.

Pause, copy the highlighted error classification and improvement into your book before moving on.

A second validation using a different relationship

We just saw how to classify and address experimental errors. That raises a question: is there an independent check that confirms the horizontal and vertical motions are truly independent? This card answers it → if $v_x = R/t$ is constant across all heights, it directly confirms the independence principle.

The theoretical time of flight is $t = \sqrt{2h/g}$. If you can measure $t$ independently (e.g., with a motion sensor or slow-motion video), you can validate:

$$R = v_x \cdot t$$

Rearranging: $v_x = R/t$. Calculate $v_x$ from each independently measured $(R,t)$ pair. Values consistent within uncertainty support the constant-horizontal-speed prediction over the tested range; they do not prove universal independence.

Theoretical times use $t = \sqrt{2h/g}$ with $g = 9.8\ \text{m/s}^2$.

Independent check: compute $v_x=R/t$ using measured $R$ and measured $t$. Consistent values within uncertainty support the constant-$v_x$ prediction. Using theoretical $t$ would not be an independent validation.

Add the highlighted reliability-check method to your notes before the activities.