Question

A lunar lander is making its descent to Moon Base I (\(\textbf{Fig. E2.40}\)). The lander descends slowly under the retro-thrust of its descent engine. The engine is cut off when the lander is 5.0 m above the surface and has a downward speed of 0.8 m/s.With the engine off, the lander is in free fall. What is the speed of the lander just before it touches the surface? The acceleration due to gravity on the moon is 1.6 m/s\(^{2}\).

Short answer

The speed of the lander just before touching the surface is approximately 4.08 m/s.

Step-by-step solution

Identify Known Values

First, let's identify the given values in the problem. The initial downward speed of the lander is 0.8 m/s (denoted as \( v_i\)). It descends from a height of 5.0 m above the surface (distance \( d = 5.0 \text{ m} \)). The gravitational acceleration on the moon is 1.6 m/s\(^2\).

Choose an Appropriate Equation

To find the final speed just before the lander touches the surface, we will use the kinematic equation:\[ v_f^2 = v_i^2 + 2ad \]where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is the acceleration, and \( d \) is the displacement.

Substitute Known Values

Substitute the known values into the kinematic equation: \( v_i = 0.8 \text{ m/s} \), \( a = 1.6 \text{ m/s}^2 \), and \( d = 5.0 \text{ m} \).Plugging them into the equation:\[ v_f^2 = (0.8)^2 + 2 \times 1.6 \times 5.0 \]

Calculate

Calculate the expression from Step 3: First calculate \( (0.8)^2 \): \( (0.8)^2 = 0.64 \)Then calculate \( 2 \times 1.6 \times 5.0 \):\( 2 \times 1.6 \times 5.0 = 16.0 \).Add these together:\[ v_f^2 = 0.64 + 16.0 = 16.64 \]

Solve for Final Velocity

To find \( v_f \), take the square root of \( v_f^2 \):\[ v_f = \sqrt{16.64} \approx 4.08 \text{ m/s} \]

Check Units and Answer

Ensure the calculated speed is in meters per second, which it is. The final speed of the lander just before it touches the surface of the moon is approximately 4.08 m/s.

Key concepts

Lunar Descent Dynamics

Lunar descent dynamics involves understanding how a spacecraft, such as a lunar lander, behaves as it approaches the lunar surface. This process includes modifications in speed and direction due to gravitational forces and engine thrust. During the landing phase:

• The lander often uses retro-thrust engines to slow down during its descent.
• Once the engines are cut off, the lander experiences free fall under the influence of lunar gravity.

The challenge is to manage these transitions smoothly to ensure a safe landing. Understanding the descent dynamics is crucial for ensuring the spacecraft lands at a manageable velocity to prevent damage or accidents.

Gravitational Acceleration on the Moon

Gravitational acceleration on the moon is significantly weaker than on Earth. It is a crucial factor in determining how objects will fall and the speed they will reach during free fall. The moon's gravity:

• Is approximately 1/6th of Earth's gravitational pull, measured at about 1.6 m/s².
• Affects the trajectory and speed of any object in free fall, such as the lunar lander in our scenario.

This lighter gravitational pull means that objects fall slower on the moon than they would on Earth. In a practical scenario like our lunar descent, it is this gravitational constant that dictates how quickly the lander speeds up once engine thrust is cut off and it begins its free fall to the lunar surface.

Kinematic Equations

Kinematic equations are a set of mathematical expressions used to describe motion. These equations are essential in predicting the future positions and velocities of moving objects under constant acceleration, like our lunar lander.The primary equation used here is:\[v_f^2 = v_i^2 + 2ad\]Where:

• \(v_f\) is the final velocity we want to determine.
• \(v_i\) is the initial velocity, given as 0.8 m/s in the problem.
• \(a\) stands for acceleration, specifically the moon's gravitational acceleration of 1.6 m/s² in this scenario.
• \(d\) denotes displacement, which is the height of 5.0 m the lander falls.

By substituting the known values into this equation, we efficiently calculate the final speed of the lander when it reaches the lunar surface. Understanding how to use kinematic equations is essential for analyzing any movement involving uniform acceleration, such as our lunar lander's free-fall scenario.