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.
1360 m/s correct answer.
(b) If an object is fired directly upward from the surface of Eris with one third 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 need help with onle part B.
To solve part B, we need to find the maximum height above the surface that an object fired upward from the surface of Eris will reach, given that its initial velocity is one-third of the escape speed.
First, let's find the escape speed from the surface of Eris, which we can use to determine the initial velocity of the object.
Escape speed (vₑ) is given by the formula:
vₑ = √(2 * g * R)
Where g is the acceleration due to gravity on the surface of Eris and R is the radius of Eris.
From the information given, we have g = 0.77 m/s² and R = 1200 km = 1,200,000 m.
Substituting these values into the formula, we can calculate the escape speed:
vₑ = √(2 * 0.77 m/s² * 1,200,000 m)
Calculating this, we find that vₑ ≈ 1359.69 m/s, which we'll round to 1360 m/s.
Now that we know the escape speed, we can determine the initial velocity of the object fired upward, which is one-third of the escape speed:
v₀ = (1/3) * 1360 m/s
v₀ ≈ 453.33 m/s, which we'll round to 453 m/s.
To find the maximum height reached by the object, we can use the conservation of energy principle.
At the maximum height, all of the kinetic energy of the object will have been converted into potential energy. Therefore, we can equate the initial kinetic energy to the final potential energy.
The initial kinetic energy of the object is given by:
KE₀ = (1/2) * m * v₀²
Where m is the mass of the object and v₀ is its initial velocity.
Since the mass of the object cancels out in the equation, we can ignore it for this calculation.
The final potential energy at the maximum height is given by:
PE_f = m * g * h
Where h is the maximum height above the surface.
Equating the initial kinetic energy to the final potential energy, we have:
(1/2) * v₀² = g * h
Solving for h, we get:
h = (1/2) * (v₀² / g)
Substituting the known values, we can calculate the maximum height:
h ≈ (1/2) * (453 m/s)² / 0.77 m/s²
Simplifying this calculation, we find that h ≈ 131,259 m.
Therefore, the object will rise to a maximum height of approximately 131,259 meters above the surface of Eris.