While visiting Planet Physics, you toss a rock straight up at 11 m/s and catch it 2.5 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 230 min. What are the (a) mass and (b) radius of Planet Physics?

Verified step by step guidance

1. Step 1: Analyze the motion of the rock. The rock is tossed straight up with an initial velocity of 11 m/s and caught after 2.5 s. Use the kinematic equation for vertical motion under constant acceleration: $y=v$₀t+12at². Here, $y$ is the displacement (which is zero since the rock returns to the starting point), $v$₀ is the initial velocity, $t$ is the time, and $a$ is the acceleration due to gravity on Planet Physics. Solve for $a$.
2. Step 2: Use the orbital motion of the cruise ship to determine the mass of Planet Physics. The cruise ship orbits at an altitude equal to the planet's radius, and the orbital period is 230 minutes. Convert the period to seconds and use the formula for orbital velocity: $v=\frac{2}{\pi }\frac{r}{T}$, where $r$ is the radius of the planet and $T$ is the orbital period. Then, use the centripetal force equation $F=\frac{\mathrm{mv}}{2}$ and equate it to the gravitational force $F=\frac{\mathrm{GMm}}{r}$ to solve for the mass of the planet.
3. Step 3: Relate the acceleration due to gravity on the surface of Planet Physics to the planet's mass and radius using the formula $g=\frac{\mathrm{GM}}{r}$. Substitute the value of $g$ (calculated in Step 1) and solve for the radius $r$.
4. Step 4: Combine the results from Step 2 and Step 3 to verify consistency. Ensure that the calculated mass and radius satisfy both the orbital motion and surface gravity conditions.
5. Step 5: Summarize the process and highlight the relationships between the rock's motion, the cruise ship's orbit, and the planet's physical properties. This reinforces the connection between kinematics, orbital mechanics, and gravitational principles.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In this scenario, the initial velocity of the rock (11 m/s) and the time of flight (2.5 s) can be used to analyze its motion using equations of motion, particularly those that relate displacement, velocity, and acceleration.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. The mass of Planet Physics can be determined by analyzing the motion of the rock and applying the gravitational acceleration formula, which relates the gravitational force to the mass of the planet and the distance from its center.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. The cruise ship's orbital period (230 min) provides information about the radius of the planet, as the relationship between the orbital radius and period can be derived from Kepler's laws and Newton's law of gravitation, allowing for the calculation of the planet's radius.