a spacecraft is launched from earth toward the moon. where will the spacecraft be when the gravitational…

Explanation:

Step1: Recall gravitational - force formula

The gravitational force between two objects is given by $F = G\frac{Mm}{r^{2}}$, where $G$ is the gravitational constant, $M$ and $m$ are the masses of the two objects, and $r$ is the distance between their centers of mass. Let the mass of the Earth be $M_E$, the mass of the moon be $M_m$, the distance between the Earth and the moon be $d$, and the distance of the spacecraft from the Earth be $x$, so the distance from the moon is $d - x$. When the gravitational forces exerted by the Earth and the moon on the spacecraft are equal, $G\frac{M_E m}{x^{2}}=G\frac{M_m m}{(d - x)^{2}}$. Canceling out $G$ and $m$, we get $\frac{M_E}{x^{2}}=\frac{M_m}{(d - x)^{2}}$.

Step2: Analyze the relationship between mass and distance

Since $M_E>M_m$, from $\frac{M_E}{x^{2}}=\frac{M_m}{(d - x)^{2}}$, we can cross - multiply to get $M_E(d - x)^{2}=M_mx^{2}$. Taking the square root of both sides and considering the physical situation, we know that for the gravitational forces to be equal, the spacecraft must be closer to the object with the smaller mass. Since the mass of the moon is smaller than the mass of the Earth, the spacecraft will be closer to the moon.

Answer:

closer to the moon than to Earth because Earth has greater mass than the moon