Introduction
In this problem, we are given a heavy particle suspended by a string of length l. The particle is given a horizontal velocity v0 and the string becomes slack at some angle, causing the particle to proceed as a projectile. We need to find the value of v0 such that the particle passes the point of suspension.

Understanding the Situation
When the string becomes slack, the particle will continue to move horizontally due to its initial velocity v0. At the same time, the force of gravity will act vertically downwards on the particle, causing it to follow a curved path. The particle will reach its maximum height at the highest point of its trajectory and then fall back down.

Analysis
To find the value of v0, we need to analyze the motion of the particle. Let's break down the problem into two stages:

1. Before the string becomes slack:
- The particle moves horizontally with a constant velocity v0.
- The tension in the string provides the necessary centripetal force for the circular motion.
- The tension T can be calculated using the formula T = mv0^2 / r, where m is the mass of the particle and r is the radius of the circular path (length of the string).
- At this stage, the particle is not a projectile yet.

2. After the string becomes slack:
- The particle continues to move horizontally due to its initial velocity v0.
- The string no longer provides any force on the particle.
- The only force acting on the particle is the force of gravity, which causes it to follow a parabolic trajectory.
- The particle becomes a projectile at this stage.

Calculating the Value of v0
To find the value of v0 so that the particle passes the point of suspension, we need to consider the highest point of the trajectory. At the highest point, the vertical component of velocity becomes zero, and the particle is momentarily at rest. Using the conservation of mechanical energy, we can equate the initial kinetic energy to the potential energy at the highest point:

1/2 mv0^2 = mgh

Where h is the maximum height reached by the particle. We can calculate h using the formula for the maximum height of a projectile:

h = (v0^2 sin^2θ) / (2g)

Where θ is the angle at which the string becomes slack.

By substituting the value of h in the first equation and solving for v0, we can find the required value of v0.

Conclusion
By considering the motion of the particle before and after the string becomes slack, we can calculate the value of v0 so that the particle passes the point of suspension. The conservation of mechanical energy and the formulas for centripetal force and maximum height of a projectile are used to analyze and solve the problem.

Its velocity should be something in between (2gl)^1/2 or (5gl)^1/2