Consider a ball of mass m, attached to a massless rod of known length, subjected to a time-dependent torque. If the initial velocity of the mass is known, then the final velocity of the mass for time t can be determined using the principle of angular impulse and momentum.

Initially, a free-body diagram of the system is drawn to illustrate all the forces acting upon the system, providing a crucial understanding of the dynamics at play. Then, the principle of angular impulse and momentum is applied to the system. This principle dictates that the initial angular momentum of an object, added to the sum of all angular impulses exerted on it over a specific period, equates to its final angular momentum.

As a result, the initial and final angular momenta of the sphere are determined by multiplying the sphere's mass, the moment arm (the perpendicular distance from the axis of rotation to the line of action of the force), and the initial and final velocities, respectively.

Next, the angular impulse is calculated by taking the integral of the torque over the given time interval. The integral can be solved by substituting the limits of the time interval into the equation. Finally, all necessary values are substituted into the initial and final angular momenta equations. By rearranging these equations, the final velocity of the sphere can be calculated.