Question

An electromagnetic rail accelerator is used to launch a research probe vertically from the surface of the Moon. The probe reaches a height of \(4.905 \mathrm{~km}\) above the surface of the Moon. What was the initial speed of the probe? Assume that the radius of the Moon is \(1737 \mathrm{~km}\) and the mass of the Moon is \(7.348 \cdot 10^{22} \mathrm{~kg}\).

Short answer

Answer: The initial speed of the probe can be found using the conservation of mechanical energy principle and the provided information. Following the steps outlined above, the initial speed, \(v\), can be calculated using the formula: \(v = \sqrt{2\left(G_\text{Moon}\frac{M_\text{Moon}m}{R} - G_\text{Moon}\frac{M_\text{Moon}m}{R+f}\right)}\).

Step-by-step solution

Calculate the gravitational constant of the Moon

First, we need to find the gravitational constant of the Moon. We will use the formula for gravitational force: \(F = GMm/r^2\), where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(7.348 \cdot 10^{22}\,\text{kg}\) is the mass of the Moon, and \(r\) is the distance between the Moon and the probe. At the surface of the Moon, \(r=R_\text{Moon}\), and we have: \[F=G\frac{M_\text{Moon}m}{R_\text{Moon}^2}\] Now, we will solve for the gravitational constant \(G\): \[G_\text{Moon}=F\frac{R^2_\text{Moon}}{M_\text{Moon}m}\]

Find the potential energy at the maximum height

When the probe reaches its maximum height, we can calculate its potential energy relative to its initial position with the formula: \[PE = -G_\text{Moon}\frac{M_\text{Moon}m}{R+f}\], where \(R = 1737 \times 10^3\, \text{m}\) is the radius of the Moon and \(f = 4.905 \times 10^3\, \text{m}\) is the height above the Moon surface.

Calculate initial mechanical energy from initial speed

When the probe is initially launched, its mechanical energy can be broken down into kinetic energy and potential energy. At this point, the probe's kinetic energy is given by the formula: \[KE = \frac{1}{2}mv^2\], where \(v\) is the initial speed we aim to find. The initial potential energy can be calculated using the same formula as in Step 2, but with \(r = R_\text{Moon}\): \[PE_\text{initial} = -G_\text{Moon} \frac{M_\text{Moon}m}{R}\] Thus, the initial mechanical energy is the sum of the initial kinetic and potential energies: \[E_\text{initial} = KE + PE_\text{initial}\].

Apply conservation of mechanical energy principle

Since mechanical energy is conserved, we can equate the total energy at the initial state and at the maximum height: \[E_\text{initial} = E_\text{max height}\] Substituting the expressions for energy, we have: \[\frac{1}{2}mv^2 - G_\text{Moon}\frac{M_\text{Moon}m}{R} = - G_\text{Moon}\frac{M_\text{Moon}m}{R+f}\]

Solve for initial speed

Now we can solve for the initial speed, \(v\), of the research probe: \[v = \sqrt{2\left(G_\text{Moon}\frac{M_\text{Moon}m}{R} - G_\text{Moon}\frac{M_\text{Moon}m}{R+f}\right)}\] Plug in the given values for \(M_\text{Moon}\), \(R\), and \(f\), as well as the value of \(G\) calculated in step 1, and compute the initial speed \(v\) of the research probe.