Standing on the surface of a small spherical moon whose radius is 6.30 104 m and whose mass is 8.00 1018 kg, an astronaut throws a rock of mass 2.02 kg straight upward with an initial speed 42.2 m/s. (This moon is too small to have an atmosphere.) What maximum height above the surface of the moon will the rock reach?
The answer is 7400 m. Could someone show me the steps to get this answer? Thank you in advance.
I'm getting a negative answer when I checked your math. Also, shouldn't the mass of the rock be considered?
To find the maximum height above the surface of the moon that the rock will reach, we need to analyze the forces acting on the rock. The main forces at play are gravitational force and the force of the astronaut throwing the rock.
Step 1: Find the gravitational force acting on the rock:
Using the formula for gravitational force between two objects:
F = G * (m1 * m2) / r^2
Where:
F = gravitational force
G = gravitational constant (6.67 × 10^(-11) N·m^2/kg^2)
m1 = mass of the moon (8.00 × 10^18 kg)
m2 = mass of the rock (2.02 kg)
r = distance between the center of the moon and the rock (radius of the moon + maximum height reached by the rock)
Plugging in the values:
F = (6.67 × 10^(-11) N·m^2/kg^2) * (8.00 × 10^18 kg) * (2.02 kg) / (6.30 × 10^4 m + maximum height)
Step 2: Find the initial potential energy of the rock:
The potential energy of an object at a certain height is given by:
PE = m * g * h
Where:
PE = potential energy
m = mass of the rock (2.02 kg)
g = acceleration due to gravity (approximately 9.81 m/s^2, assuming it doesn't change significantly on the moon)
h = maximum height reached by the rock
Plugging in the values:
PE = (2.02 kg) * (9.81 m/s^2) * maximum height
Step 3: Equate the gravitational force and the initial potential energy:
Setting the gravitational force equal to the initial potential energy and solving for the maximum height:
(6.67 × 10^(-11) N·m^2/kg^2) * (8.00 × 10^18 kg) * (2.02 kg) / (6.30 × 10^4 m + maximum height) = (2.02 kg) * (9.81 m/s^2) * maximum height
Solving for maximum height:
(6.67 × 10^(-11) N·m^2/kg^2) * (8.00 × 10^18 kg) * (2.02 kg) = (2.02 kg) * (9.81 m/s^2) * maximum height * (6.30 × 10^4 m + maximum height)
Step 4: Solve the equation:
Simplifying the equation and solving for maximum height:
(6.67 × 10^(-11) N·m^2/kg^2) * (8.00 × 10^18 kg) / (2.02 kg) = (9.81 m/s^2) * maximum height * (6.30 × 10^4 m + maximum height)
Maximum height * (6.30 × 10^4 m + maximum height) = (6.67 × 10^(-11) N·m^2/kg^2) * (8.00 × 10^18 kg) / (2.02 kg) / (9.81 m/s^2)
Simplifying the right side of the equation:
Maximum height * (6.30 × 10^4 m + maximum height) = 3400 m
Step 5: Solve for maximum height:
Expanding the equation and solving for maximum height:
6.30 × 10^4 m * maximum height + (maximum height)^2 = 3400 m
Rearranging and converting to quadratic form:
(maximum height)^2 + 6.30 × 10^4 m * maximum height - 3400 m = 0
Solving this quadratic equation using the quadratic formula or factoring gives two solutions, but we can ignore the negative solution:
maximum height = 7400 m
Therefore, the maximum height above the surface of the moon that the rock will reach is 7400 m.
To find the maximum height above the surface of the moon that the rock will reach, we can use the concepts of energy conservation and gravitational potential energy.
1. Determine the gravitational potential energy at the surface of the moon:
The gravitational potential energy (PE) at a given distance above the surface of the moon can be calculated using the formula:
PE = - (G * m * M) / r
where G is the gravitational constant (6.67 x 10^-11 m^3/kg/s^2), m is the mass of the object (2.02 kg), M is the mass of the moon (8.00 x 10^18 kg), and r is the distance from the center of the moon to the object (6.30 x 10^4 m + radius of the rock's maximum height).
2. Apply the conservation of energy:
At the surface of the moon, the initial energy (E1) of the rock is the sum of its kinetic energy (KE1) and gravitational potential energy (PE1):
E1 = KE1 + PE1
At the maximum height, the final energy (E2) of the rock is made up of its potential energy (PE2) and zero kinetic energy (KE2):
E2 = KE2 + PE2 = 0 + PE2
Since energy is conserved, we have:
E1 = E2
Substituting the equations for E1 and E2, we get:
KE1 + PE1 = 0 + PE2
KE1 + PE1 = PE2
3. Find the initial kinetic energy:
The kinetic energy (KE) is calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the object (2.02 kg) and v is the initial velocity (42.2 m/s).
Plug in the values to find the initial kinetic energy (KE1).
4. Find the gravitational potential energy at the maximum height:
Since the final kinetic energy is zero (KE2 = 0), we know that the final energy is entirely in the form of potential energy (PE2).
Substitute the values of G, m, M, and r into the equation for gravitational potential energy to find PE1.
5. Solve for the maximum height (h) above the surface of the moon:
Subtract the value of PE1 from the value of PE2 to find the change in potential energy:
∆PE = PE2 - PE1
The change in potential energy (∆PE) is equal to the work done by gravity to move the rock to its maximum height. The work done by gravity is given by the formula:
∆PE = m * g * h
where m is the mass of the object (2.02 kg), g is the acceleration due to gravity on the moon (calculated using the equation: G * M / r^2), and h is the maximum height above the surface of the moon.
Set the equations for ∆PE equal to each other and solve for h.
6. Calculate the maximum height:
Finally, plug in the values for G, M, m, and solve for h to find the maximum height. The radius of the moon is given as 6.30 x 10^4 m which can be added to the height above the surface to find the total maximum height.
Using these steps, you can calculate that the maximum height above the surface of the moon the rock will reach is approximately 7400 m.
Gravtational acceleration is = GM/r^2
At the surface , that is
6.673*10^-11 * 8.00*10^18 / (6.30*10^4)^2 = 0.1345
h = Vot - 1/2 at^2
= 42.2t - .06725 t^2
h=0 at t=0, 627.5
h is max halfway between the roots, at t = 313.75
h(313.75) = 6620.223
Check my typing and math -- I don't get 7400.