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.

Sample data (for comparison if you cannot perform the experiment):

These sample data assume $v_x \approx 1.0\ \text{m/s}$ and $g = 9.8\ \text{m/s}^2$.

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). Draw a line of best fit.

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 $\sqrt{h}$; gradient $m_\text{exp} = \Delta R/\Delta\sqrt{h}$; theoretical gradient $m_\text{theory} = v_x\sqrt{2/g}$; accept $<10\%$ difference. Launch speed from data: $v_x = m_\text{exp}\sqrt{g/2}$.

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 errors:

• Parallax in height measurement: Read the ruler at eye level.
• Uncertainty in landing position: The ball may bounce or roll slightly. Carbon paper helps mark the first contact point.
• Timing uncertainty: If measuring time of flight with a stopwatch, human reaction time (~0.2 s) is a significant source of error.

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.

Systematic errors (tilted ramp, air resistance, inconsistent release) shift all results in one direction; random errors (parallax, landing spread) are reduced by averaging. Key improvement: photogate at ramp exit directly 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 $(R, t)$ pair. If $v_x$ is approximately constant across all heights, this confirms that horizontal speed is unaffected by vertical motion — a fundamental assumption of the projectile model.

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

Reliability check: compute $v_x = R/t$ at each height (using theoretical $t = \sqrt{2h/g}$). A constant $v_x$ across all heights confirms that horizontal speed is unaffected by vertical motion — the key assumption of the projectile model.

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