Problem 14 - Basic Technical Mathematics with Calculus

Question

The engine of a lunar lander is cut off when the lander is \(5.0 \mathrm{m}\) above the surface of the moon and descending at \(2.0 \mathrm{m} / \mathrm{s}\). If the acceleration due to gravity on the moon is \(1.6 \mathrm{m} / \mathrm{s}^{2},\) what is the speed of the lander just before it touches the surface?

Step-by-Step Solution

Answer

The speed of the lander just before it touches the surface is approximately 4.47 m/s.

Step 1: Identify Known Values

We know the initial velocity of the lunar lander is \( v_0 = -2.0 \, \text{m/s} \) (negative because it is descending), the initial position is \( y_0 = 5.0 \, \text{m} \), and the acceleration due to gravity is \( a = -1.6 \, \text{m/s}^2 \) (also negative, as it acts in the same direction as the velocity towards the moon's surface).

Step 2: Use Kinematic Equation to Find Final Velocity

We use the kinematic equation \( v^2 = v_0^2 + 2a(y - y_0) \). The final position \( y \) is the surface of the moon, so \( y = 0 \) (taking the surface as the reference point).

Step 3: Substitute Values into Kinematic Equation

Substitute the known values into the equation: \[ v^2 = (-2.0)^2 + 2(-1.6)(0 - 5.0) \]Simplify to find:\[ v^2 = 4 + 16 = 20 \]

Step 4: Solve for Final Velocity

Take the square root to solve for the final velocity \( v \):\[ v = \sqrt{20} = 4.47 \, \text{m/s} \]Since the lander is still descending, the velocity is \(-4.47 \, \text{m/s} \). However, speed is the magnitude, so it is not negative.

Key Concepts

Lunar Module

Lunar modules, or landers, are specially designed spacecraft used during moon missions. They are crucial components engineered to transport astronauts and equipment to and from the moon’s surface safely. A lunar lander, such as the one described in the problem, must manage various phases of its mission, including descent, landing, and ascent from the lunar surface.

During descent, the lunar module utilizes its engines to slow down and control its fall to the moon’s surface. The engine is crucial to counteract the moon’s gravitational pull and to provide a safe landing speed. Once the engine is cut off, the module relies on gravity alone, making it important to calculate speeds and distances accurately.

This exercise illustrates the moment when the module’s engine ceases operation, and the lander freefalls under the influence of lunar gravity. The computation involved helps in understanding how speed changes during this freefall, which is essential for ensuring a precise and safe landing.

Acceleration Due to Gravity

Acceleration due to gravity on the moon is significantly less than on Earth. On the moon, gravity measures about one-sixth of Earth's gravitational pull. Specifically, it is approximately 1.6 m/s². This means objects on the moon accelerate downward more slowly than they would on Earth.

In kinematics problems like the lunar module descent, gravity is a key factor that affects the velocity and acceleration of descending objects. The gravitational constant helps predict how swiftly the lander will speed up or slow down as it approaches the moon's surface. Since both gravity and initial velocity act in the same direction toward the surface, they are both considered negative in the exercise.

Understanding the acceleration due to gravity is crucial for planning safe lunar landings and other physical activities conducted on the moon.

Initial Velocity

The initial velocity of an object is the speed and direction it has at the beginning of the observation period. For the lunar module in this exercise, the initial velocity is 2 m/s downward, which is given as -2.0 m/s. The negative sign here indicates the direction is towards the surface.

Initial velocity is an essential parameter in kinematics. It forms the starting point for calculating further motion changes when forces such as gravity act on it. In the case of our lunar module, the negative velocity signifies that the module is descending or moving downward.

Recognizing the initial velocity allows for the application of kinematic equations to determine other critical motion characteristics, such as final velocity or displacement over time. It is a fundamental step in solving many physics problems.

Kinematic Equations

Kinematic equations form the backbone of motion analysis for objects under uniform acceleration. They enable us to calculate various aspects of motion such as velocity, displacement, and time. In the problem, the specific kinematic equation used is:

\[ v^2 = v_0^2 + 2a(y - y_0) \]

This equation relates the final velocity \( v \), initial velocity \( v_0 \), acceleration \( a \), and displacement \( y - y_0 \). When using these equations, having the correct signs for directions is important. Here, both acceleration and initial velocity have negative signs because they point towards the moon's surface.

Using kinematic equations is a systematic way to predict future positions and velocities, critical in planning aerospace missions. This framework for motion is vital for understanding how objects move under the influence of constant forces like gravity.