The escape velocity of a planet depends on its mass and radius. If the diameter of the Earth is increased by 4 times, it means the radius will also increase by 2 times.
Let's assume the original radius of the Earth is r, and the mass of the Earth remains unchanged. The new radius after the increase will be 2r.
The formula for escape velocity is:
v = sqrt((2 * G * M) / r)
where: v = escape velocity G = gravitational constant M = mass of the Earth r = radius of the Earth
If we substitute the new radius (2r) into the formula, we get:
v' = sqrt((2 * G * M) / (2r))
Since the mass of the Earth (M) remains unchanged, we can simplify the equation further:
v' = sqrt((G * M) / r)
As you can see, the expression for the escape velocity remains the same. Therefore, if the Earth's diameter is increased 4 times, the escape velocity would remain unchanged.