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A projectile is launched from the foot of a fixed incline that rises…

Question

A projectile is launched from the foot of a fixed incline that rises at 30 degrees to the horizontal. The launch speed is 20 m/s and the launch direction (in the vertical plane containing the line of greatest slope) is chosen to maximize the distance measured along the incline surface from the launch point to the landing point. Taking g = 10 m/s^2, find this maximum range along the incline, in metres.

✓ Verified answer: 26.6667checked by our engine — not a guess

Step-by-step solution

For a projectile launched up a plane inclined at angle alpha, the range along the incline is R = 2u^2 cos(theta) sin(theta - alpha) / (g cos^2 alpha), where theta is the launch angle from horizontal.

Maximizing over theta gives optimal theta = (90 + alpha)/2 (the launch direction bisects the incline and the vertical).

The maximum range simplifies to R_max = u^2 / [g(1 + sin alpha)].
With u = 20, alpha = 30 deg (sin alpha = 1/2), g = 10: R_max = 400 / [10(1 + 0.5)] = 400/15 = 26.6667 m.

Final answer26.6667

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