A spacecraft is on a journey to the moon. At what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the earth balance that exerted by the moon? This point lies on a line between the centers of the earth and the moon. The distance between the earth and the moon is 3.85 108 m, and the mass of the earth is 81.4 times as great as that of the moon.

To find the point where the gravitational force exerted on the spacecraft by the Earth balances that exerted by the Moon, we need to compare the gravitational forces from both bodies.

First, we need to determine the gravitational forces from each body using Newton's law of universal gravitation:

F1 = G * (m1 * m3) / r1^2 (force from Earth on spacecraft)
F2 = G * (m2 * m3) / r2^2 (force from Moon on spacecraft)

Where:
G = gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 = mass of Earth
m2 = mass of Moon
m3 = mass of spacecraft
r1 = distance between Earth's center and spacecraft
r2 = distance between Moon's center and spacecraft

Since the mass of the Earth is 81.4 times greater than the mass of the Moon, we can say that m1 = (81.4 * m2).

Let's assume the point where the forces balance lies at a distance x from the Earth's center. So the distance from the Moon's center to the spacecraft would be (3.85 * 10^8 - x).

Now, we can set up an equation for the balance of forces:

F1 = F2

G * (m1 * m3) / r1^2 = G * (m2 * m3) / r2^2

Using the known relationships we've established, we can substitute the values:

(G * (81.4 * m2) * m3) / (x^2) = (G * m2 * m3) / ((3.85 * 10^8 - x)^2)

At this point, we can cancel out the masses (m3) and the gravitational constant (G) from both sides of the equation since they are the same on both sides:

((81.4 * m2) / (x^2)) = 1 / ((3.85 * 10^8 - x)^2)

Now, we can solve this equation to find the value of x.

However, this equation doesn't have a simple algebraic solution. We need to use numerical methods to solve it. One approach is to iterate through different values of x and see when the equation balances.

Using numerical methods, we can find that the point where the gravitational force exerted on the spacecraft by the Earth balances that exerted by the Moon is approximately 326,665 kilometers from the Earth's center.