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 and effect of gravity considering short distance firing)
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.