To calculate the new escape velocity, we can use the equation for escape velocity:
Ve = sqrt((2GM)/R)
Where: Ve 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 celestial body (in this case, Earth). R is the radius of the celestial body (in this case, Earth).
Given that the escape velocity of Earth is 11.2 km/s, we can convert it to meters per second:
Ve = 11.2 km/s = 11,200 m/s
Now, let's calculate the new escape velocity by considering the changes in mass and radius:
New mass of Earth (M') = 2 * current mass of Earth (M) New radius of Earth (R') = (1/8) * current radius of Earth (R)
Using these values, we can calculate the new escape velocity (Ve'):
Ve' = sqrt((2G * M') / R')
Let's plug in the values and calculate:
M' = 2 * M R' = (1/8) * R
Ve' = sqrt((2G * 2M) / ((1/8) * R))
Simplifying further:
Ve' = sqrt(16 * (2G * M) / R)
Since G, M, and R remain constant in this equation, we can simplify it to:
Ve' = 4 * Ve
Therefore, the new escape velocity (Ve') will be four times the original escape velocity (Ve):
Ve' = 4 * 11,200 m/s = 44,800 m/s
So, if Earth's mass increases to twice its present value and the radius of Earth becomes one eighth, the new escape velocity will be 44,800 m/s.