What is the ratio of the escape velcoities from two planets of equal densities but different masses `M_(1)` and `M_(2)` ?

Explanation

To find the ratio of escape velocities from two planets of equal densities but different masses \( M_1 \) and \( M_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Escape Velocity Formula**:
The escape velocity \( v \) from a planet is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Relate Density to Mass and Volume**:
Since the planets have equal densities, we can express the mass in terms of density and volume. The density \( \rho \) is given by:
\[ \rho = \frac{M}{V} \]
The volume \( V \) of a planet can be expressed as:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, we can write:
\[ M = \rho V = \rho \left(\frac{4}{3} \pi R^3\right) \]

3. **Express Radius in Terms of Mass**:
Rearranging the equation for mass, we get:
\[ R^3 = \frac{3M}{4\pi\rho} \]
Taking the cube root gives:
\[ R = \left(\frac{3M}{4\pi\rho}\right)^{1/3} \]

4. **Substitute Radius into Escape Velocity Formula**:
Now substitute \( R \) back into the escape velocity formula:
\[ v = \sqrt{\frac{2GM}{\left(\frac{3M}{4\pi\rho}\right)^{1/3}}} \]
Simplifying this gives:
\[ v = \sqrt{\frac{2G M}{\left(\frac{3M}{4\pi\rho}\right)^{1/3}}} = \sqrt{\frac{2G M \cdot (4\pi\rho)^{1/3}}{(3M)^{1/3}}} \]

5. **Simplify the Escape Velocity Expression**:
This can be further simplified to:
\[ v \propto M^{1/3} \]
Therefore, we can conclude that the escape velocity is proportional to the cube root of the mass of the planet.

6. **Calculate the Ratio of Escape Velocities**:
Let the escape velocities for planets 1 and 2 be \( v_1 \) and \( v_2 \) respectively. Then:
\[ \frac{v_1}{v_2} = \sqrt{\frac{M_1}{M_2}} \]

### Final Answer:
The ratio of the escape velocities from two planets of equal densities but different masses \( M_1 \) and \( M_2 \) is:
\[ \frac{v_1}{v_2} = \sqrt{\frac{M_1}{M_2}} \]