The velocity of the center of gravity of a system depends on the individual velocities of the objects and their masses. To determine the velocity of the center of gravity when an object moves with velocity X and a person standing at the end of that object moves with velocity Y in the opposite direction, we need to consider the masses and velocities of both objects.
Let's assume the object has mass M₁ and velocity X, and the person has mass M₂ and velocity Y. The center of gravity velocity (V_cg) can be calculated using the formula:
V_cg = (M₁ * V₁ + M₂ * V₂) / (M₁ + M₂)
where V₁ is the velocity of the object and V₂ is the velocity of the person.
If the object and the person are considered as a combined system, the total mass (M_total) would be the sum of their individual masses:
M_total = M₁ + M₂
Now, substituting the given values into the formula, we can find the velocity of the center of gravity:
V_cg = (M₁ * X + M₂ * (-Y)) / (M₁ + M₂)
The negative sign in front of Y accounts for the opposite direction of motion of the person.
Please note that the direction of the velocity will depend on the signs and relative magnitudes of X and Y.