More practice questions

4. Q4. A ball rolls off a table 1.25 m high at 3 m s^⁻¹. How far from the foot of the table does it land (g = 10 m s^⁻²)?

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   Fall time from 1.25 = ½ × 10 × t^²: t = 0.5 s. Horizontal distance = 3 × 0.5 = 1.5 m.

5. Q5. For an upward launch with 0° < θ₀ < 90°, returning to the same level without air resistance, at what angle is the range four times the maximum rise?

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   v₀² sin 2θ₀/g = 4 × v₀² sin^² θ₀/2g gives 2 sin θ₀ cos θ₀ = 2 sin^² θ₀, so tan θ₀ = 1 and θ₀ = 45°.

6. Q6. Two balls are projected with the same speed at 30° and 60°. Compare their ranges, maximum heights and times of flight.

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   Ranges are equal (sin 60° = sin 120°). Heights are in the ratio sin^² 30° : sin^² 60° = 1 : 3. Times of flight are in the ratio sin 30° : sin 60° = 1 : √3.

7. Q7. A ball is projected at 20 m s^⁻¹ at 60° above the horizontal. When is its velocity perpendicular to its initial velocity (g = 10 m s^⁻²)?

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   v · v₀ = v₀² − gtv₀ sin θ₀ = 0 gives t = v₀/(g sin θ₀) = 20/(10 × 0.866) ≈ 2.31 s, before it lands at T_f ≈ 3.46 s. For a nonvertical upward throw landing at its launch height, this occurs strictly before landing only if θ₀ > 45°; at 45° it occurs exactly at landing.