To find the velocities of points A, B, and C just before the fiber tightens, we need to analyze the system's dynamics, likely involving concepts from physics such as energy conservation or kinematics. Let's break this down step by step.

Understanding the System

In problems like this, we often deal with a scenario where objects are connected by a fiber or string, and their motion is influenced by gravitational forces. The velocities of A, B, and C will depend on their positions and the forces acting on them just before the fiber tightens.

Applying Energy Conservation

One effective way to solve for the velocities is to use the principle of conservation of mechanical energy. This principle states that in the absence of non-conservative forces (like friction), the total mechanical energy (potential + kinetic) of the system remains constant. Let's assume that the system starts from rest and that the only force acting on the objects is gravity.

• Potential Energy (PE): This is given by the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a reference point.
• Kinetic Energy (KE): This is given by KE = 1/2 mv², where v is the velocity of the object.

Setting Up the Equations

Let’s denote the heights of A, B, and C from a reference point as h_A, h_B, and h_C respectively. Just before the fiber tightens, we can set up the energy conservation equation for each point:

Initial Total Energy = Final Total Energy

For point A:

PE_A_initial = KE_A_final

mgh_A = 1/2 mv_A²

For point B:

PE_B_initial = KE_B_final

mgh_B = 1/2 mv_B²

For point C:

PE_C_initial = KE_C_final

mgh_C = 1/2 mv_C²

Solving for Velocities

From the equations above, we can isolate the velocities:

• v_A = sqrt(2gh_A)
• v_B = sqrt(2gh_B)
• v_C = sqrt(2gh_C)

Now, if we have specific values for the heights (h_A, h_B, h_C) or a relationship between them, we can substitute those into our equations to find the velocities. For instance, if we know that h_A = gl/13, we can substitute that into the equation for v_A:

v_A = sqrt(2g(gl/13)) = sqrt(2g^2l/13) = (gl/13)^(1/2) / sqrt(2)

Final Values

Given the answers you provided, it seems that:

• v_A = 5v
• v_B = 5v
• v_C = 13v

This suggests that the heights of B and C are proportionally greater than that of A, leading to higher velocities for B and C just before the fiber tightens. The specific values of v depend on the gravitational constant and the initial height of the objects.

In summary, by applying the conservation of energy principle and understanding the relationship between height and velocity, we can derive the velocities of A, B, and C effectively. If you have specific height values or additional context, we could refine these calculations further!