Eris, the largest dwarf planet known in the Solar System, has a radius R = 1200 km and an acceleration due to gravity on its surface of magnitude g = 0.77 m/s2.
(a) Use these numbers to calculate the escape speed from the surface of Eris.
(b) If an object is fired directly upward from the surface of Eris with one fourth of this escape speed, to what maximum height above the surface will the object rise? (Assume that Eris has no atmosphere and negligible rotation.)
I know the answer to a) is 1360 m/s, but I cannot figure out b). I know I should be using the conversation equation. However, I don't know how to apply it.
Solution to a similar problem has been posted earlier which please check.
B) The total mechanical energy = KE + PE of the object at the surface of Eris will remain constant and will be equal to PE at the max. height h (above the surface). The object's KE will be zero at the max. ht. attained.
To solve part (b) of the problem, we can use the conservation of energy principle. The initial kinetic energy of the object can be related to its final potential energy at the maximum height.
Let's assume the object has mass m.
1. Start by calculating the escape speed from part (a) as follows:
Escape speed (Ve) = sqrt(2gR)
Substituting the given values, we have:
Ve = sqrt(2 * 0.77 m/s^2 * 1200 km * 1000 m/km)
Note that we convert the radius from km to meters by multiplying by 1000.
Calculating Ve gives us Ve ≈ 1360 m/s.
2. Now, let's consider the object fired upward with one fourth (1/4) of the escape speed.
Initial kinetic energy (KE_i) = (1/2) * m * (1/4)^2 * Ve^2
= (1/32) * m * Ve^2
At the maximum height, the object's velocity becomes zero, and all its initial kinetic energy is converted into potential energy. Therefore:
Final potential energy (PE_f) = m * g * h
Where h is the maximum height above the surface.
3. Applying the conservation of energy principle:
KE_i = PE_f
(1/32) * m * Ve^2 = m * g * h
Simplifying and canceling out the mass:
(1/32) * Ve^2 = g * h
Rearranging the equation:
h = (1/32) * (Ve^2 / g)
4. Plugging in the values:
h = (1/32) * (1360 m/s)^2 / 0.77 m/s^2
Simplifying:
h ≈ 2941 meters
Therefore, the object will rise to a maximum height of approximately 2941 meters above the surface of Eris when fired directly upward with one fourth of the escape speed.