The equation to find the velocity at which an object reaches its terminal velocity is given by the drag equation, which describes the force of drag experienced by an object as it moves through a fluid. The drag equation is given by the following formula:

F_d = 1/2 * C_d * A * p * v^2

Where F_d is the force of drag, C_d is the drag coefficient, A is the cross-sectional area of the object, p is the density of the fluid, and v is the velocity of the object.

The drag equation is derived by considering the balance of forces acting on an object as it moves through a fluid. In particular, the drag force is proportional to the velocity squared and the other factors in the equation. As the object reaches its terminal velocity, the drag force is equal to the gravitational force acting on the object, and it can no longer continue to accelerate.

To find the terminal velocity, we set the net force acting on the object to zero and solve for velocity. This gives us the following equation for terminal velocity:

v_t = sqrt((2 * m * g) / (C_d * A * p))

Where m is the mass of the object and g is the acceleration due to gravity. This equation shows that the terminal velocity depends on the mass, size, and shape of the object, as well as the density of the fluid it is moving through.

In conclusion, the equation to find the velocity at which an object reaches its terminal velocity is the drag equation. This equation is derived by considering the balance of forces acting on an object as it moves through a fluid and setting the net force to zero. The terminal velocity is determined by the mass, size, and shape of the object, as well as the density of the fluid it is moving through.
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