Final answer:
The speed of the particle at its maximum height is zero. At this point, all the initial kinetic energy has been transformed into gravitational potential energy, resulting in a speed of zero.
Step-by-step explanation:
The speed of the particle at its maximum height in vertical projectile motion will be zero. When a particle attached to a string is given an initial velocity in the vertical direction, it rises until it reaches a point where its vertical velocity component is zero; this point is the maximum height. At the maximum height, the particle has no kinetic energy because its speed is zero. Gravitational potential energy is at its maximum because it is directly proportional to the height of the particle from the ground. Therefore, any other form of energy, such as kinetic energy implied by velocity, is zero, as it has been fully converted into potential energy.
It is important to understand that at the maximum height, the vertical component of a projectile's velocity is always zero, regardless of the mass of the particle or the stiffness of the system. This stems from the principle of conservation of energy, which indicates that the initial kinetic energy is transformed into potential energy completely at the peak of the trajectory.