Practical Investigation: Validating Projectile Motion

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 $v_x\sqrt{2/g}$.

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.

Trial	Height $h$ (m)	$\sqrt{h}$ (m½)	Range $R_1$ (m)	Range $R_2$ (m)	Range $R_3$ (m)	Mean Range $\bar{R}$ (m)	Uncertainty (m)
1
2
3
4
5

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

Height $h$ (m)	$\sqrt{h}$ (m½)	Mean Range $\bar{R}$ (m)
0.10	0.316	0.32
0.20	0.447	0.45
0.30	0.548	0.55
0.40	0.632	0.64
0.50	0.707	0.71

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

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).

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.

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.

Height $h$ (m)	Theoretical $t$ (s)	Measured $t$ (s)	$\%$ difference
0.20	0.20
0.40	0.29
0.60	0.35

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