Practical Investigation: Validating Projectile Motion

Q2 — Variables

Independent variable (IV): launch (exit) height h, measured from the floor to the centre of the ball at the ramp exit. Varied deliberately across five values from 0.100 m to 0.500 m.

Dependent variable (DV): horizontal range R, measured from the point on the floor directly below the ramp exit to the centre of the landing mark.

Controlled variables (any three of):

• Launch speed v_x: the ball is released from rest at the same marked position on the ramp every trial, so the gravitational PE → KE conversion gives the same exit speed each time.
• The ball: a single dense steel ball bearing is used for every trial to keep mass, diameter and air-resistance behaviour constant.
• Launch angle: the ramp exit is checked horizontal with a spirit level before each set of trials, so the launch angle is consistently 0°.
• Surface and air conditions: the same lab bench and floor surface are used in still air, so rolling friction and air movement do not vary between trials.

Marking note: 1 mark IV, 1 mark DV, 1 mark per properly identified controlled variable up to 3.

Q3 — Plot & transformation table (expected values)

Plotted points (√h, R̄): (0.316, 0.32), (0.447, 0.45), (0.548, 0.55), (0.632, 0.64), (0.707, 0.71). Error bar lengths (with Δh = 0.005 m and ΔR̄ = 0.02 m): vertical ±0.02 m on every point; horizontal Δ√h = Δh / (2√h) = e.g. 0.005 / (2 × 0.316) = 0.0079 m^½ at h = 0.100 m, shrinking to 0.0035 m^½ at h = 0.500 m.

The data lie almost exactly on a straight line through the origin. Line of best fit:

• m_exp ≈ 1.00 m^½ (gradient by least-squares through origin: Σ(xy) / Σ(x²) = 1.286 / 1.286 = 1.000).
• c ≈ 0 (intercept passes through origin within error, as the model predicts — this itself is part of the validation).
• Implied v_x = m_exp √(g/2) = 1.00 m^½ × √4.90 m s⁻² = 1.00 × 2.214 = 2.21 m s⁻¹.

Units check on v_x: (m^½) × √(m s⁻²) = (m^½) × (m^½ s⁻¹) = m s⁻¹ ✓.

Q4 — Uncertainty propagation (h = 0.300 ± 0.005 m, R̄ = 0.55 ± 0.02 m)

(a) Central value.

v_x = R̄ √(g / 2h) = (0.55 m) × √((9.80 m s⁻²) / (2 × 0.300 m)) = (0.55 m) × √(16.33 s⁻²) = (0.55 m) × (4.041 s⁻¹) = 2.223 m s⁻¹.

(b) Fractional uncertainties.

ΔR̄ / R̄ = 0.02 / 0.55 = 0.0364 (3.64%).

½ · Δh / h = 0.5 × (0.005 / 0.300) = 0.5 × 0.01667 = 0.00833 (0.83%).

(c) Combine in quadrature.

(Δv_x / v_x)² = (0.0364)² + (0.00833)² = 0.001325 + 0.0000694 = 0.001394.

Δv_x / v_x = √0.001394 = 0.0373 (3.73%).

Δv_x = 0.0373 × 2.223 m s⁻¹ = 0.0830 m s⁻¹.

(d) Quote with consistent rounding.

Round Δv_x to 1 sig fig: Δv_x ≈ 0.08 m s⁻¹. Round central value to the same decimal place: v_x = 2.22 m s⁻¹.

v_x = (2.22 ± 0.08) m s⁻¹.

Marking note: the dominant uncertainty contribution comes from ΔR̄ (3.64%), not Δh (only 0.83% after the ½ factor). This justifies the experimental priority on accurate range measurement (carbon paper + plumb line) over even finer height measurement.

Q5 — Sample band 5–6 extended response (annotated)

[Mark 1 — testable prediction] The experimental design isolates one specific testable prediction of the projectile model: with v_x held constant (controlled by releasing the ball from the same marked release line on the ramp) and g taken as 9.80 m s⁻², the horizontal range R should depend on launch height h according to R = v_x√(2h/g), which linearises to R proportional to √h with theoretical gradient v_x√(2/g) and zero intercept.

[Mark 2 — refers to graph] The R vs √h plot in Section 3 yields a line of best fit through the origin with gradient m_exp ≈ 1.00 m^½ and intercept c ≈ 0. Both the linearity and the zero intercept directly support the model: a non-linear plot or non-zero intercept would falsify the √h dependence.

[Mark 3 — uses uncertainty quantitatively] The implied launch speed from the gradient is v_x = (2.22 ± 0.08) m s⁻¹ (from Section 4 propagation of ±0.005 m in h and ±0.02 m in R̄). This 3.7% relative uncertainty is well inside the lesson's <10% acceptance threshold, providing quantitative validation rather than mere visual agreement.

[Mark 4 — random vs systematic error addressed by design] Random error in range — caused by small inconsistencies in release, ball spin and landing-mark spread — is reduced by averaging three trials at each height and quantified by the ±0.02 m range uncertainty used in error bars and uncertainty propagation. Systematic error, in contrast, would shift every measurement in the same direction and is addressed by (i) checking the ramp exit is horizontal with a spirit level (eliminating a non-zero launch angle), (ii) using a plumb line to locate the point directly below the exit (eliminating a constant offset in R), and (iii) using a dense steel ball bearing to keep air resistance negligible.

[Mark 5 — quantitative judgement] Because (a) the data are linear in √h, (b) the intercept is consistent with zero, and (c) the implied v_x is determined to ~4% precision, the model R = v_x√(2h/g) is validated within the resolution of the experiment.

[Mark 6 — residual error identified] Residual error not removed by the design includes rolling friction along the ramp, which slowly bleeds kinetic energy and very slightly reduces v_x in the higher-PE (taller release-line) configuration; if significant, this would produce a small downward curvature in the R vs √h plot at high h, which is not visible in the present data but would become detectable with a wider range of h.

[Mark 7 — improvement / reliability tie-in] An obvious extension to tighten the test is to insert a photogate at the ramp exit to measure v_x directly, then compare m_exp√(g/2) with the photogate v_x as an independent cross-check — exactly the strategy raised in the lesson's Reliability Check (Card 6) and Improvements list (Card 5).

Marking notes by band:

• Band 3 (1–2 marks): identifies the formula being tested; describes the experiment in narrative form without using the graph or uncertainty calculation.
• Band 4 (3–4 marks): uses the linearised graph as evidence; mentions random and systematic error but does not quantify them.
• Band 5 (5–6 marks): brings the quantitative result v_x = (2.22 ± 0.08) m s⁻¹ from Section 4 into the judgement; correctly distinguishes random from systematic error and ties each to a specific design feature.
• Band 6 (7 marks): all of the above, plus identifies a plausible residual error not removed by the design and proposes a targeted improvement (e.g. photogate cross-check) that directly addresses it.