Problem 15 - Chemical Principles: The Quest for Insight

Question

Mars orbits the \(5 u n\) at \(25 \mathrm{~km} / \mathrm{s}^{-1}\). A spuceship attempting to land on Mars must match its orbital speed. If the mass of the spaceship is \(2.5 \times 10^{5} \mathrm{~kg}\), what is its kinetic energy when its speed has matched that of Mars?

Step-by-Step Solution

Answer

The spaceship's kinetic energy when matching Mars' orbital speed is \(7.8125 \times 10^{11} \mathrm{J}\).

Step 1: Understand the concept of kinetic energy

Kinetic energy (KE) is the energy that an object has due to its motion. It is given by the formula KE = \(\frac{1}{2}mv^2\), where 'm' is the mass of the object and 'v' is the velocity of the object.

Step 2: Identify the known variables

The mass of the spaceship (m) is given as \(2.5 \times 10^5 \mathrm{kg}\) and the orbital speed to match Mars (v) is \(25\mathrm{km/s}\). Convert the speed to meters per second by multiplying by 1000, which gives \(25000\mathrm{m/s}\).

Step 3: Calculate the kinetic energy of the spaceship

Use the kinetic energy formula with the given mass and the converted speed to calculate the kinetic energy: \(KE = \frac{1}{2}mv^2 = \frac{1}{2}(2.5 \times 10^5\mathrm{kg})(25000\mathrm{m/s})^2\).

Step 4: Perform the calculation

Substituting the values into the equation, we get \(KE = \frac{1}{2}(2.5 \times 10^5\mathrm{kg})(625 \times 10^6\mathrm{m^2/s^2})\). Multiplying these, KE = \(0.5 \times 2.5 \times 625 \times 10^5 \times 10^6\mathrm{J}\) which equals \(7.8125 \times 10^{11}\mathrm{J}\).

Key Concepts

Kinetic Energy Formula

Understanding the kinetic energy formula is fundamental in physics because it describes the energy an object possesses due to its motion. The general formula for kinetic energy (KE) is expressed as \( KE = \frac{1}{2}mv^2 \), where \( m \) represents the mass of the object and \( v \) its velocity.

To get a feel for this concept, consider a baseball thrown through the air. The ball has kinetic energy, which depends on how heavy the ball is (its mass) and how fast it is thrown (its velocity). As either the mass or velocity increases, so does the kinetic energy. For example, a bowling ball (\

Orbital Speed

Orbital speed is the velocity at which an object must travel to maintain a stable orbit around another object, due to the balance of gravitational force and inertial motion. In the case of a spaceship needing to land on Mars, it must match the orbital speed of Mars to enter its orbit without being pushed away or pulled apart.

It's like riding a merry-go-round. To stay on without being thrown off or collapsing towards the center, you need to maintain a certain speed. This is the essence of orbital speed in the context of celestial bodies. The exercise concerns a practical application of physics, as matching orbital speeds is a crucial step for any spacecraft aiming to rendezvous with a planet or orbital station.

Physics Problem Solving

Tackling physics problems effectively involves a series of strategic steps aimed at breaking down the complexities into manageable parts. The initial step is to understand the principles involved—in our example, kinetic energy and orbital speeds. Next, identify all the known quantities and their units, ensuring they are in compatible units suitable for the calculations.

Subsequently, apply the relevant formulas and perform the necessary computations. In our scenario, after determining the mass of the spaceship and its required orbital speed, we can input these into the kinetic energy formula. The systematic approach to problem-solving is vital for learning and applying physics in practical scenarios, as it ensures that the foundational concepts are understood and the calculations are executed precisely.