IB Physics Projectile Thrown from a Cliff

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The maximum range of a projectile on the horizontal level of its projection (neglecting air resistance) occurs when the angle of projection to the horizontal is 45°. A more difficult problem is when the point of projection is above the horizontal plane on which the object lands. A tutorial sheet of difficult “show that” problems is given below.

  1. A projectile is thrown at an angle 𝜽 to the horizontal at a speed u from a vertical height h. Show that the horizontal range of the projectile is u2/(2g)( sin2𝜽 +√(sin22𝜽 +8ghcos2𝜽/u2) ).
  2. A projectile is thrown at a speed u from a height h above horizontal ground. Show that the angle of projection to the horizontal for maximum range on the ground is tan-1√ (u2/(u2+2gh) ).
  3. A projectile is thrown at a speed u from a height h above horizontal ground. Show that the maximum range of the projectile is u/g√(u2+2gh).
  4. A projectile is thrown from a vertical height above the ground and maximum range occurs when the angle of projection to the horizontal is 𝜽. Show that the angle (in the maximum range case only) at which the projectile strkes the horizontal is 90° - 𝜽.