If the distance is halved, the gravitational force increases by a factor of 4, so the new force is 40 N.

The unit of gravitational potential is Joules per kilogram (J/kg).

The gravitational potential at infinity is defined to be 0.

A negative gravitational potential energy indicates that the masses are attracting each other.

The gravitational potential energy U at height h is given by U = mgh.

The gravitational potential energy U of an object in a gravitational field is given by U = -GMm/r.

The gravitational potential energy U = -G(m1*m2)/r.

The half-life of a first-order reaction is given by t1/2 = 0.693/k, which depends only on the rate constant.

The heat transfer coefficient in convection depends on surface area, fluid velocity, and temperature difference.

Q1 = 3, Q3 = 6. Interquartile Range = Q3 - Q1 = 6 - 3 = 3.

The Langmuir isotherm assumes that adsorption sites are equivalent and that the process is monolayer.

The Langmuir isotherm assumes that adsorption occurs on a surface with a finite number of equivalent sites.

The maximum number of molecules that can be adsorbed at a given temperature is referred to as the saturation capacity.

Minimum possible length = Measured value - Error = 50 - 1 = 49 cm.

Since 7^2 + 24^2 = 25^2, triangle ABC is a right triangle.

To determine the nature of the lines, we can analyze the discriminant of the quadratic equation.

The lines are perpendicular if the product of their slopes is -1. We can find the slopes from the equation and check this condition.

The discriminant is negative, indicating that the lines are perpendicular.

The lines intersect at the origin (0,0) as derived from the equation.

The lines intersect at the origin, which can be verified by substituting x = 0 and y = 0 into the equation.

The lines are perpendicular if the product of their slopes is -1, which can be verified from the equation.

The lines intersect at the origin and are not parallel, hence they are intersecting.

Completing the square shows that the lines intersect at two distinct points.

The equation can be factored as (x - 3y)^2 = 0, indicating that the lines are coincident.

The magnetic field inside a long solenoid is uniform and parallel to the axis of the solenoid.

To find the maximum, we calculate f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. Since f''(x) = -2 < 0, this is a maximum point.

The vertex form of a parabola gives the maximum value at x = -b/(2a) = 2. Evaluating f(2) = -2^2 + 4*2 + 1 = 9.

The vertex of the parabola given by f(x) = -x^2 + 4x + 1 occurs at x = -b/(2a) = -4/(-2) = 2, which gives the maximum value.

The mean free path is the average distance a molecule travels before colliding with another molecule.

Coefficient of Variation = (Standard Deviation / Mean) * 100 = (5 / 50) * 100 = 10%.