There are several ways to get this type of motion with a motor. One is to use a large gear reduction to generate the amount of torque you need to get the motion you want. Another way is to convert the rotary motion of the motor into linear motion with a lead screw for example, as shown in
this video for a robotic arm. We used a lead screw with a PG71 gearmotor to adjust the angle of our shooter last year. A rack and pinion type of mechanism may also achieve the motion you are looking for. Many times a counterbalance is applicable to take some of the load off of the motor, increase speed, and/or increase reliability. This could be a type of spring, surgical tubing, gas shocks, or any other way that energy can be stored to counteract gravity or other external forces that the motor will need to overcome to tilt the shooter.
Our team is heavily considering pneumatic actuation as a solution to tilt our shooter because we believe that having a known position without sensor, set point, or mechanical error will be beneficial for consistent shooting, and our strategy does not rely on being able to have a continuously variable shooting angle. A position for starting configuration and another for shooting is enough.