The escape velocity of a bullet from the surface of planet Y is 1951.0 m/s. Calculate the escape velocity from the surface of the planet X if the mass of planet X is 1.35 times that of Y, and its radius is 0.871 times the radius of Y.
To calculate the escape velocity from the surface of planet X, we need to use the formula for escape velocity:
\(v_e = \sqrt{\frac{{2GM}}{{R}}}\)
Where:
- \(v_e\) is the escape velocity
- \(G\) is the universal gravitational constant (\(6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}\))
- \(M\) is the mass of the planet
- \(R\) is the radius of the planet
Let's start by finding the mass and radius of planet X:
Given:
Mass of planet Y = \(M_Y\)
Mass of planet X = \(1.35 \times M_Y\)
Radius of planet Y = \(R_Y\)
Radius of planet X = \(0.871 \times R_Y\)
Now, we can substitute the values into the escape velocity formula for planet X:
\(v_{eX} = \sqrt{\frac{{2G(1.35 \times M_Y)}}{{0.871 \times R_Y}}}\)
Since we know the escape velocity of planet Y is 1951.0 m/s, we can set up the following equation:
\(1951.0 = \sqrt{\frac{{2G \times M_Y}}{R_Y}}\)
Now we just need to solve for \(M_Y\) and \(R_Y\) to find the escape velocity from the surface of planet X.
To do this, we can square both sides of the equation:
\(1951.0^2 = \frac{{2G \times M_Y}}{{R_Y}}\)
Rearranging the equation:
\(M_Y = \frac{{1951.0^2 \times R_Y}}{{2G}}\)
Now we can substitute this expression for \(M_Y\) back into the equation for the escape velocity of planet X:
\(v_{eX} = \sqrt{\frac{{2G(1.35 \times \frac{{1951.0^2 \times R_Y}}{{2G}})}}{{0.871 \times R_Y}}} = \sqrt{{1.35 \times 1951.0^2 \times \frac{{R_Y}}{{0.871 \times 2G}}}}\)
Finally, we can calculate the escape velocity from the surface of planet X by plugging in the values:
\(v_{eX} = \sqrt{{1.35 \times 1951.0^2 \times \frac{{R_Y}}{{0.871 \times 2 \times 6.67430 \times 10^{-11}}}}}\)
Make sure to substitute the values for \(R_Y\) and solve the equation to find the final result.