Question

(a) Calculate the escape speed on Europa, a satellite of the planet Jupiter. The radius of Europa is \(1569 \mathrm{~km}\) and the freefall acceleration at its surface is \(1.30 \mathrm{~m} / \mathrm{s}^{2} .(b)\) How high will a particle rise if it leaves the surface of the satellite with a vertical velocity of \(1.01 \mathrm{~km} / \mathrm{s} ?(c)\) With what speed will an object hit the satellite if it is dropped from a height of \(1000 \mathrm{~km} ?\) ( \(d\) ) Calculate the mass of Europa.

Short answer

Following the steps in order, the escape speed, maximum height reached, impact speed, and mass of Europa should be calculated accordingly.

Step-by-step solution

Calculate the Escape speed

The escape speed from any celestial body can be calculated using the formula: \(v_e = \sqrt{2gr}\), where \(g\) is the acceleration due to gravity, and \(r\) is the radius of the body. Here, \(g = 1.3 m/s^2\) and \(r = 1569 km = 1569000 m\). By inserting these into the formula, the escape velocity is computed.

Determine the maximum height

The maximum height an object will reach when launched with an initial velocity can be calculated using the equation \(h = v^2/2g\), where \(v\) is the initial launch velocity and \(g\) is the acceleration due to gravity. Here, the initial speed is given as \(1.01 km/s = 1010 m/s\). Inserting these values into the equation, the maximum height can be determined.

Calculate the impact speed

The impact velocity of an object can be calculated using the formula \(v = \sqrt{2gh}\), where \(g\) is the acceleration due to gravity and \(h\) is the height from which the object is dropped. Here, \(h = 1000 km = 1000000 m\). This equation yields the impact velocity on the surface of Europa.

Calculate the mass

The mass of Europa can be inferred from the surface freefall acceleration using the formula \(m = gr^2 / G\), where \(g\) is the acceleration due to gravity, \(r\) is the radius of the planet and \(G\) the gravitational constant. Substituting \(g = 1.30 m/s^2\), \(r = 1569 km = 1569000 m\) and \(G = 6.67430 × 10^{-11} m^3 ⋅ kg^{-1} ⋅ s^{-2}\) into the equation, the mass of Europa can be calculated.

Key concepts

Escape Speed

When we talk about escape speed, we're discussing the minimum velocity which an object must have to break free from a celestial body’s gravitational influence and move into space. For celestial bodies like moons, planets, and stars, this speed depends on their mass and radius. Physicists use a formula to determine the escape velocity, which is expressed as \( v_e = \sqrt{2gr} \), where \( g \) represents the gravitational acceleration on the surface of the body, and \( r \) its radius.

In the context of the exercise, to escape Europa, which is one of Jupiter's moons, one would need to reach the calculated speed. The higher the gravitational pull (related to the mass and size of the celestial body), the greater the escape speed required. It's a crucial concept for space missions, as it dictates the minimum energy required for a spacecraft to leave a planet or moon's surface and start its journey into space.

Freefall Acceleration

The term freefall acceleration, denoted as \( g \), refers to the acceleration experienced by an object solely under the influence of gravity, with no other forces acting on it. To determine how fast an object would accelerate if dropped from a height, or the force of gravity at a given location, this acceleration is crucial. On Earth, we have a standard acceleration due to gravity equal to about \( 9.81 m/s^2 \).

However, other celestial bodies have different freefall accelerations. The exercise shows how we use \( g \) to calculate other quantities, such as escape speed and impact velocity. It also plays a significant role in the mass calculation of a celestial body, indicating that even without leaving the surface, we can infer a lot about a celestial body's mass and density.

Gravitational Force

The concept of gravitational force is at the heart of celestial mechanics. It's the force that attracts any two objects with mass. The more massive an object is, the more significant its gravitational attraction. Isaac Newton's Law of Universal Gravitation quantifies this force and shows that two objects attract each other with a force that's directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The formula for gravitational force is \( F = G \frac{{m_1 m_2}}{{r^2}} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) the masses of the objects, and \( r \) the distance between their centers. Gravitational force determines the orbits of planets, the tides on Earth, and the trajectories necessary for space exploration.

Celestial Body Mass Calculation

Estimating the mass of a celestial body is a fundamental aspect of understanding its gravitational influence. In the exercise, we learn how to calculate the mass of Europa by using the formula \( m = \frac{{gr^2}}{{G}} \). Here, \( g \) again is the freefall acceleration or the surface gravity of Europa, \( r \) is Europa’s radius, and \( G \) the gravitational constant—known to be about \( 6.67430 \times 10^{-11} m^3 \cdot kg^{-1} \cdot s^{-2} \).

Knowing the mass of a celestial body like Europa allows us to understand more about its composition and structure. It also helps in planning missions, navigating spacecraft, and exploring potential resources. By using physics to our advantage, we can estimate these masses even when direct measurements are not possible.