We present QUaRTM - a novel quadcopter design capable of tilting the propellers into the forward flight direction, which reduces the drag area and therefore allows for faster, more agile, and more efficient flight. The vehicle can morph between two configurations in mid-air, including the untilted configuration and the tilted configuration. The vehicle in the untilted configuration has a higher pitch torque capacity and a smaller vertical dimension. The vehicle in the tilted configuration has a lower drag area, leading to a higher top speed, higher agility at high speed, and better flight efficiency. The morphing is accomplished without any additional actuators beyond the four motors of a quadcopter. The rigid connections between the quadcopter frame and the quadcopter arms are replaced with sprung hinges. This allows the propellers to be tilted when high thrusts are produced, and recover to the untilted configuration when the thrusts are brought low. The effectiveness of such a vehicle is demonstrated by running experiments on a prototype vehicle with a shape similar to a regular quadcopter. Through the use of tilting, the vehicle is shown to have a 12.5% higher maximum speed, better high-speed agility as the maximum crash-free cruise speed increased by 7.5%, and a better flight efficiency as the power consumption has dropped by more than 20% in the speed range of 15-20 m s-1.