Final answer:
On a planet four times as massive as Earth and the same size, the gravitational acceleration would be about 39.24 m/s². Using the formula for gravitational potential energy, the energy needed to lift a 2 kg mass by 2 m on this planet would be 156.96 Joules.
Step-by-step explanation:
Energy Needed to Lift a Mass on a Heavy Earth-like Planet
To calculate the energy needed to lift a 2 kg mass vertically upward through a 2 m distance on a planet with four times the mass of Earth but the same size, we must first consider the gravitational force on that planet. Since the mass is quadrupled, and the radius is the same, the gravitational acceleration would be four times greater than that on Earth (assuming uniform density). On Earth, gravitational acceleration is approximately 9.81 m/s² (which we often round to 10 m/s² for simplicity). On this hypothetical planet, it would be roughly 4 times that, so about 39.24 m/s².
The formula for calculating gravitational potential energy (G.P.E.) is G.P.E. = mgh, where 'm' is mass, 'g' is the acceleration due to gravity, and 'h' is the height. Substituting the given values, we have G.P.E. = 2 kg * 39.24 m/s² * 2 m, which equals 156.96 Joules. Therefore, the energy required to lift the 2 kg mass the specified distance on this planet is 156.96 Joules.