I apologize for the confusion in my previous response. You are correct that if the velocity increases, the displacement over time will also increase. Let's consider the scenario again and calculate the correct values.
Given: Original escape velocity, Ve = 11.2 km/s = 11,200 m/s Mass of Earth, M Radius of Earth, R
New mass of Earth (M') = 2 * M New radius of Earth (R') = (1/8) * R
To calculate the new escape velocity (Ve'), we'll use the equation:
Ve' = sqrt((2GM')/R')
Substituting the new values:
Ve' = sqrt((2G * 2M) / ((1/8) * R))
Simplifying further:
Ve' = sqrt((4G * M) / ((1/8) * R))
Ve' = sqrt(32G * M / R)
Therefore, the correct new escape velocity (Ve') will be sqrt(32) times the original escape velocity (Ve):
Ve' = sqrt(32) * Ve
Calculating this value:
Ve' = sqrt(32) * 11,200 m/s ≈ 19,781 m/s
Thus, if Earth's mass increases to twice its present value and the radius of Earth becomes one eighth, the new escape velocity will be approximately 19,781 m/s.