A ball is launched from ground level at speed (18.0 ± 0.3) m s⁻¹ and angle (35.0 ± 1.0)° above horizontal. Ignore drag and use g = 9.81 m s⁻². Calculate time of flight, maximum height and range from the central values. Estimate the percentage uncertainty in range using R = u²sin(2θ)/g and maximum-uncertainty addition, treating angular uncertainty in radians. Explain two reasons a measured range could be systematically shorter.

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Time = 2.10 s, maximum height = 5.43 m, range = 31.0 m. Estimated range uncertainty ≈ 4.6%, or about ±1.4 m. Drag and a launch point below the assumed level can shorten the range. Resolve velocity: u_y = 18.0sin35° = 10.32 m s⁻¹ and u_x = 18.0cos35° = 14.74 m s⁻¹. For equal launch and landing heights, T = 2u_y/g = 2.10 s. Maximum height is u_y²/(2g) = 5.43 m. Range is u_xT, equivalently u²sin70°/g = 31.0 m. These two routes provide an algebraic cross-check. For R ∝ u²sin2θ, fractional maximum uncertainty is approximately 2Δu/u + |2cot(2θ)|Δθ = 2(0.3/18.0) + 2cot70°(π/180) = 0.0460. Thus ΔR ≈ 1.4 m. Air drag reduces horizontal speed; a landing surface below or above the assumed equal level changes flight time. A launch-speed calibration bias or angle reference offset would also shift all trials systematically.

Projectile prediction with uncertainty

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