Imagine a spaceship on its way to the moon from the earth. Find the point, as measured from the center of the earth, where the force of gravity due to the earth is balanced exactly by the gravity of 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.

help anyone

hey are you from city college

Use newton's law of gravitation

F=G*(m1m2/r^2)
we get:
G*(81.4 m1*m2)/r^2 = G*(m1*m2)/(3.85x10^8-r)^2

you can then cancel the G, m1, m2 and get
81.4/r^2=1/(3.85x10^8-r)^2

take sqrt:
sqrt(81.4)(3.85x10^8-r)= r

solve for r
r=346,585,261m

..... and yes Tenzin hes from CCNY lol

Hey, hedberg! I have him right now for Spring 2016...save me.

i mean hes a wonderful teacher. He just goes crazy hard on tests and quizzes. gonna fail lmfao

울산 알바몬(052.albamon.com) | 잡코리아가 만든 아르바이트 No.1

To find the point, as measured from the center of the earth, where the force of gravity due to the earth is balanced exactly by the gravity of the moon, we can use the concept of gravitational force.

Gravitational force between two objects depends on the mass of the objects and the distance between them. The force of gravity between the earth and the moon can be calculated using Newton's law of universal gravitation:

F = G * (m1 * m2) / r²

Where:
F is the gravitational force,
G is the gravitational constant (6.67430 x 10^-11 N(m/kg)^2),
m1 and m2 are the masses of the interacting bodies (earth and moon, respectively),
and r is the distance between the centers of the two bodies.

Given that the distance between the earth and the moon is 3.85 x 10^8 m, we can calculate the force of gravity between them.

However, we know that at the point where the forces of gravity from the earth and moon balance each other, the net gravitational force should be zero. So, we equate the gravitational forces due to the earth and the moon:

G * (m1 * m2) / r1² = G * (m1 * m2) / r2²

Where r1 and r2 are the distances between the center of the earth and the point in question, and the center of the moon and the point in question, respectively.

Given that the mass of the earth is 81.4 times greater than the mass of the moon, we can substitute m1 = 81.4m2 into the equation:

G * (81.4m2 * m2) / r1² = G * (m2 * m2) / r2²

Now we can simplify the equation:

81.4 / r1² = 1 / r2²

To find the point where the forces balance, we need to find the ratio of the distances r1 and r2.

By cross-multiplying and simplifying further, we get:

(r1/r2)² = 81.4

Taking the square root of both sides gives us:

r1/r2 = √81.4 = 9.03

So, the distance r1 is 9.03 times larger than r2, that is:

r1 = 9.03 * r2

Given that the distance between the earth and the moon is 3.85 × 10^8 m, we can set up the equation:

r1 + r2 = 3.85 × 10^8

Substituting the relationship between r1 and r2, we have:

9.03 * r2 + r2 = 3.85 × 10^8

Simplifying further:

10.03 * r2 = 3.85 × 10^8

To find r2, we divide both sides by 10.03:

r2 = (3.85 × 10^8) / 10.03

Calculating this, we find:

r2 ≈ 3.837 × 10^7 m

Finally, we can find r1 by multiplying r2 by 9.03:

r1 = 9.03 * (3.837 × 10^7)

Calculating this, we find:

r1 ≈ 3.462 × 10^8 m

Therefore, the point where the force of gravity due to the earth is balanced exactly by the gravity of the moon, as measured from the center of the earth, is approximately 3.462 × 10^8 m from the center of the earth.