Range and maximum height
Objective. Calculate key flight characteristics.
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- 2. Worked example
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Learn: the key idea
Range and maximum height are key flight characteristics of a projectile launched from ground level at speed u and angle θ: the maximum height occurs when vertical velocity is zero, giving H = (u²sin²θ)/(2g), and the horizontal range on level ground is R = (u²sin2θ)/g, derived from the time of flight T = (2u sin θ)/g. The range formula shows that, for fixed speed, the maximum range occurs at θ = 45°, since sin2θ is maximised there. For example, doubling the launch angle from 20° to 40° changes the range through the sin2θ term, not simply by doubling.
Worked example
A projectile is launched from ground level at 25 m/s at 40° to the horizontal. Calculate its maximum height, taking g = 9.8 m/s².
- 1Maximum height H = (u sin θ)² ÷ (2g)
- 2u sin θ = 25 sin 40° ≈ 16.07
- 3H = (16.07)² ÷ (2 × 9.8)
- 4H ≈ 258.2 ÷ 19.6 ≈ 13.2 m
Answer: H ≈ 13.2 m
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Start practisingCommon mistake. Students use T = (u sin θ)/g instead of the full time of flight (2u sin θ)/g, halving the range. Always use the complete time of flight, accounting for both the rise and the fall.
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