Question

Standing on the surface of a small spherical moon whose radius is \(6.30 \cdot 10^{4} \mathrm{~m}\) and whose mass is \(8.00 \cdot 10^{18} \mathrm{~kg}\), an astronaut throws a rock of mass \(2.00 \mathrm{~kg}\) straight upward with an initial speed \(40.0 \mathrm{~m} / \mathrm{s}\). (This moon is too small to have an atmosphere.) What maximum height above the surface of the moon will the rock reach?

Short answer

Solution: To find the maximum height, follow these steps: 1. Calculate the initial potential energy (PE_i) of the rock using the formula PE_i = -G * m1 * m2 / r, where G is the gravitational constant, m1 is the mass of the moon, m2 is the mass of the rock, and r is the radius of the moon. 2. Calculate the initial kinetic energy (KE_i) of the rock using the formula KE_i = (1/2) * m * v^2, where m is the mass of the rock and v is its initial speed. 3. Calculate the initial total mechanical energy (E_i) of the rock by adding the initial kinetic energy and the initial potential energy: E_i = KE_i + PE_i. 4. Set the final potential energy (PE_f) equal to the initial total mechanical energy (E_i): PE_f = E_i. 5. Solve for the final radius (r_f) by setting the final potential energy equal to the initial total mechanical energy in terms of r_f: r_f = G * m1 * m2 / -E_i. 6. Find the maximum height (h_max) above the surface of the moon by subtracting the radius of the moon (r) from the final radius (r_f): h_max = r_f - r. Following these steps will help you find the maximum height the rock reaches above the surface of the moon when thrown with an initial speed v.

Step-by-step solution

Calculate the initial potential energy of the rock

To calculate the initial potential energy of the rock, we can use the formula for gravitational potential energy, which is given by: \[ PE_{i} = -\frac{G \cdot m_{1} \cdot m_{2}}{r} \] where \(PE_{i}\) is the initial potential energy, \(G\) is the gravitational constant \((6.674 \cdot 10^{-11} \frac{\mathrm{Nm^2}}{\mathrm{kg^2}})\), \(m_{1}\) is the mass of the moon, \(m_{2}\) is the mass of the rock, and \(r\) is the radius of the moon.

Calculate the initial kinetic energy of the rock

The initial kinetic energy of the rock can be calculated using the formula for kinetic energy, which is given by: \[ KE_{i} = \frac{1}{2}mv^2 \] where \(KE_{i}\) is the initial kinetic energy, \(m\) is the mass of the rock, and \(v\) is the initial speed of the rock.

Calculate the initial total mechanical energy of the rock

The initial total mechanical energy of the rock can be calculated by adding the initial kinetic energy and the initial potential energy: \[ E_{i} = KE_{i} + PE_{i} \]

Calculate the final potential energy of the rock

At the maximum height, the rock's kinetic energy will be zero, and its total mechanical energy will be the same as its initial total mechanical energy. So, we can set the final potential energy equal to the initial total mechanical energy: \[ PE_{f} = E_{i} \]

Solve for the final radius

The final potential energy can also be expressed as a function of the final radius \(r_{f}\), which is the distance from the center of the moon to the rock at its maximum height: \[ PE_{f} = -\frac{G \cdot m_{1} \cdot m_{2}}{r_{f}} \] By setting this equal to the initial total mechanical energy, we can solve for the final radius: \[ r_{f} = \frac{G \cdot m_{1} \cdot m_{2}}{-E_{i}} \]

Find the maximum height above the surface

Finally, we can find the maximum height above the surface by subtracting the radius of the moon from the final radius: \[ h_{max} = r_{f} - r \] After calculating the initial potential energy, the initial kinetic energy, and the initial total mechanical energy, we can substitute the values and find the final radius and the maximum height the rock reaches above the surface of the moon.