To determine the angle and velocity of a catapult for a physics project, measuring the horizontal distance and time of flight can be effective methods. Using a protractor attached to the catapult or recording the launch in slow motion can provide accurate angle measurements. Calculating the initial velocity involves analyzing the height and distance traveled by the projectile. Additionally, setting up a video recording with a grid background can help trace the projectile's path for precise calculations. These techniques will aid in understanding projectile motion effectively.