The escape velocity of a bullet from the surface of planet Y is 1695 m/s. Calculate the escape velocity from the surface of the planet X if the mass of planet X is 1.37 times that of Y, and its radius is 0.887 times the radius of Y (in m/s).
To calculate the escape velocity from the surface of planet X, we can use the formula for escape velocity:
v = sqrt(2 * G * M / r),
where:
- v is the escape velocity,
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
- M is the mass of the planet, and
- r is the radius of the planet.
Given that the escape velocity from the surface of planet Y is 1695 m/s, let's denote this velocity as vy and the corresponding mass and radius of planet Y as My and ry, respectively.
Therefore, using the formula above for planet Y:
vy = sqrt(2 * G * My / ry).
Now, let's calculate the escape velocity from the surface of planet X denoted as vx, my, and rx, respectively. We are given that the mass of planet X is 1.37 times that of Y, and its radius is 0.887 times the radius of Y.
Mathematically, we can write it as:
Mx = 1.37 * My, (1)
rx = 0.887 * ry. (2)
Substituting the values into the formula above for planet X:
vx = sqrt(2 * G * Mx / rx).
From equations (1) and (2), we can substitute Mx and rx in terms of My and ry:
vx = sqrt(2 * G * (1.37 * My) / (0.887 * ry)).
vx = sqrt(2 * G * My * 1.37 / 0.887 * ry).
Now, to find the escape velocity from the surface of planet X, we can substitute the given values of vy, My, and ry into the equation above:
vx = sqrt(2 * G * 1.37 * My / 0.887 * ry).
Substituting the value of vy:
1695 = sqrt(2 * G * My / ry).
We can now solve for vy:
1695 = sqrt(2 * G * 1.37 * My / 0.887 * ry),
1695 = sqrt(2 * G * My * 1.37 / 0.887 * ry) * sqrt(1).
Taking the square of both sides:
1695^2 = (2 * G * My * 1.37 / 0.887 * ry),
2866025 = (2 * G * My * 1.37 / 0.887 * ry).
Now, rearranging the equation to solve for vx:
vx = sqrt(2 * G * My * 1.37 / 0.887 * ry).
Finally, substituting the values from equations (1) and (2):
vx = sqrt(2 * G * (1.37 * My) / (0.887 * ry)).
Thus, the escape velocity from the surface of planet X is sqrt(2 * G * (1.37 * My) / (0.887 * ry)) m/s.