In projectile motion without air resistance, two launch angles that add up to
90° (called complementary angles) will produce the
same horizontal range when launched with the same initial speed.
the range for angles such as 30° and 60° will be identical. These are called
range pairs.
But what about the height?
Although complementary angles give the same range, they do not have the same:
maximum height
flight time
trajectory shape
A smaller angle (e.g., 30°) produces a lower, flatter path with a smaller height, while
a larger angle (e.g., 60°) produces a higher trajectory with a greater maximum height.
Thus, complementary angles give the same range, but not the same height or time of flight.
Problem: Arrow Shot Towards a Wall
An arrow is shot towards a wall 50m away with an initial velocity that makes an angle of 45° with the horizontal. It strikes the wall 35m above the ground. Assuming the arrow was released from ground level, find the initial velocity of the arrow.
Problem: Projectile Fired from the Top of a Building at an Angle
A projectile is fired with a muzzle velocity of 300m/s from a gun aimed upward at an angle of 20° with the horizontal, from the top of a building 30m. high above a level ground. With what velocity will it hit the ground, in m/s?
Problem: Bullet Shot Dead Center Toward a Target
A bullet shoots at 460 m/s at a target 130 m away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center?
What is the maximum height achieved by the bullet?
Problem: Projectile Fired at a Distance from the Edge of a Cliff
50m from the edge of a cliff, a projectile is thrown at an angle of 30° with the horizontal. At what horizontal distance beyond the cliff does the ball strike the water if it just barely misses the edge of the cliff? The height of the cliff 62m.
Problem: Dropping a Load at a Horizontal Velocity
A small plane, flying at 180km/h at an altitude of 240m, is to drop an inflatable raft to flood victims stranded on a flat roof. How far from the roof should the pilot release the package so that it will land on the roof?
Problem: Ball Thrown just to Clear a Wall
A ball is thrown so that it just clears a 25-ft wall 100ft. away. If it left the hand 5 feet above the ground an at an angle of 60° to the horizontal, what was the initial velocity of the ball?
Problem: Projectile Thrown from the Edge of a Cliff
A projectile with a muzzle velocity of 550m/s is fired from a gun on top of a cliff 460m above the sea level. If the projectile hits the ocean surface 49.2 seconds after being fired, determine (a) the angle of elevation of the gun, (b) the range, and (c) the highest point attained above the cliff.
Problem: Concept of Complementary Angles Giving the Same Range
A catapult is placed 100 ft from the castle wall, which is 35 feet high. The soldier wants the burning bale of hay to clear the top of the wall and land 50 feet inside the castle wall. If the initial velocity of the bale is 70 feet per second, at what angle should the bale of hay be launched so that it travel 150 feet and pass over the castle wall. Use $g = 32 ft/sec^2$.
