A

Step 1: Understand that the gravitational field \( \mathbf{g} \) is related to the gravitational potential \( V \) by the equation:
\( \mathbf{g} = -\nabla V \)
This means that the gravitational field is the negative gradient of the potential. We can compute the gradient in terms of its components as follows:

• \( \frac{\partial V}{\partial x} = 20 \)
• \( \frac{\partial V}{\partial y} = 40 \)

Step 2: Compute the gravitational field components:
\( g_x = -\frac{\partial V}{\partial x} = -20 \) and \( g_y = -\frac{\partial V}{\partial y} = -40 \)
Therefore, the gravitational field vector is:
\( \mathbf{g} = (-20, -40) \) N/kg.
Step 3: Now calculate the magnitude of the gravitational field:
\( |\mathbf{g}| = \sqrt{(-20)^2 + (-40)^2} = \sqrt{400 + 1600} = \sqrt{2000} = 20\sqrt{5} \approx 44.72 \) N/kg.
Step 4: Now, find the gravitational force on a particle of mass 0.250 kg at the point (2, 4):
\( F = m \cdot g = 0.250 \times |\mathbf{g}| = 0.250 \times 20\sqrt{5} \approx 11.18 \) N.
Therefore, the gravitational field at point (2, 4) is: \( -20 \, \mathbf{i} - 40 \, \mathbf{j} \) N/kg and the gravitational force on the particle is approximately 11.18 N.