Final answer:
Using conservation of energy, the student's kinetic energy at the start of the swing converts to potential energy when velocity is zero. The angle at which the rope is released can be found by relating the height of the swing's arc to the angle using trigonometry, then applying the conservation of energy principle.
Step-by-step explanation:
To determine the angle θ at which the student releases the rope when his velocity is zero, we can use the principles of conservation of energy.
At the beginning of the student's swing, the only energy present is kinetic energy, given by the formula KE = ½ mv², where m is mass and v is velocity. When the student reaches the highest point of the swing and is about to release the rope, all this kinetic energy has been converted into potential energy (PE), given by PE = mgh, where g is the acceleration due to gravity and h is the height above the lowest point of the swing.
Since the student releases the rope when their velocity is zero, at this point, they have maximum potential energy and no kinetic energy. By equating the initial kinetic energy to the final potential energy (KE = PE) and factoring in the height as a function of the swing's arc length and the angle θ, we can solve for the angle.
The exact calculation will depend on the length of the rope, which is not given in the problem. However, the approach would involve calculating the height of the swing's arc at the point of release using trigonometry, then applying the conservation of energy principle to find the angle θ.