Question

An electromagnetic rail accelerator is used to launch a research probe vertically from the surface of the Moon. The initial speed of the projectile is \(114.5 \mathrm{~m} / \mathrm{s}\). What height does it reach above the surface of the Moon? 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: To find the maximum height, we use the conservation of mechanical energy and the formula for gravitational potential energy (PE) and kinetic energy (KE). According to the conservation of mechanical energy, initial KE is equal to final PE. When we set KE equal to PE and solve for height (h), we get the equation: \(h = \dfrac{v^2}{2g}\). Plugging in the provided values and calculating, we can find the maximum height the projectile reaches above the Moon's surface.

Step-by-step solution

Calculate gravitational constant on the Moon:

Using the mass and radius of the Moon, we can calculate the gravitational constant on the Moon's surface (g). This can be found using the formula: \(g = \dfrac{GM}{R^2}\), where \(G\) is the universal gravitational constant (\(6.674 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{kg}^2\)), \(M\) is the mass of the Moon, and \(R\) is the radius of the Moon. Plugging in provided values: \(g = \dfrac{6.674 × 10^{-11} \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{kg}^2 \times 7.348 × 10^{22} \mathrm{~kg}}{(1737 × 10^3 \mathrm{~m})^2}\)

Calculate the gravitational potential energy (PE):

Now we're going to calculate the gravitational potential energy when the probe is at its highest point. At this point, the probe's kinetic energy will be 0. The mechanical energy is conserved throughout the probe's journey, so the total initial energy (which is equal to the initial kinetic energy) will equal the total final energy (which is equal to the final PE). The formula for gravitational PE is: PE = mgh, where m is the mass of the probe, g is the gravitational constant on the Moon, and h is the height we are looking for.

Calculate the initial kinetic energy (KE):

To calculate the initial kinetic energy of the probe, we use the formula: KE = \(\dfrac{1}{2}mv^2\), where m is the mass of the probe and v is its initial speed. Since the mechanical energy is conserved, m will cancel out when we compare KE to PE.

Set KE equal to PE and solve for height h:

According to the conservation of mechanical energy, initial KE is equal to final PE. Therefore: \(\dfrac{1}{2}mv^2 = mgh\) We can cancel m from both sides, leaving us with: \(\dfrac{1}{2}v^2 = gh\) Now we solve for h: \(h = \dfrac{v^2}{2g}\) Plugging in provided values: \(h = \dfrac{(114.5 \mathrm{~m} / \mathrm{s})^2}{2 \times g}\) Solving for h, we get the height the probe reaches above the Moon's surface.