A dishonest carnival shop owner bends the barrel of an air rifle downward by $15^\circ$ near the middle, causing unsuspecting visitors to miss the balloon target. To compensate and successfully burst the balloon, at what approximate angle above the target should a clever visitor aim? (Neglect air resistance)
Answer
To understand how to compensate for the tampered rifle, we can break down the mechanics of the modification and the required adjustment:
The Defect: Bending the rifle barrel downward by $15^\circ$ near its middle introduces a permanent angular offset to the muzzle. When the user aligns the sights of the rifle directly with the target ($0^\circ$), the altered trajectory of the barrel causes the projectile to exit at a $15^\circ$ downward angle relative to the line of sight, resulting in a miss below the balloon.
The Compensation: To successfully hit the target, the visitor must counteract the $15^\circ$ downward deflection built into the barrel. By tilting the entire rifle upward by $15^\circ$ relative to the target, the shooter elevates the rear mechanism.
The Trajectory: This upward tilt precisely cancels out the $15^\circ$ downward bend of the barrel, returning the muzzle to a position parallel to the original horizontal line of sight. Consequently, the bullet exits moving directly toward the balloon.
Conclusion: The clever visitor must aim $15^\circ$ above the target to successfully burst the balloon.