Problem 75 - College Physics

Question

\(\bullet$$\bullet\) Mars Exploration Rover landings. In January 2004 the Mars Exploration Rover spacecraft landed on the surface of the Red Planet, where the acceleration due to gravity is 0.379 what it is on earth. The descent of this 827 kg vehicle occurred in several stages, three of which are outlined here. In Stage I, friction with the Martian atmosphere reduced the speed from \(19,300 \mathrm{km} / \mathrm{h}\) to 1600 \(\mathrm{km} / \mathrm{h}\) in a 4.0 min interval. In Stage II, a parachute reduced the speed from 1600 \(\mathrm{km} / \mathrm{h}\) to 321 \(\mathrm{km} / \mathrm{h}\) in \(94 \mathrm{s},\) and in Stage III, which lasted 2.5 \(\mathrm{s}\) , retrorockets fired to reduce the speed from 321 \(\mathrm{km} / \mathrm{h}\) to zero. As part of your solu- tion to this problem, make a free-body diagram of the rocket during each stage. Assuming constant acceleration, find the force exerted on the spacecraft (a) by the atmosphere during Stage I, (b) by the parachute during Stage II, and (c) by the retrorockets during Stage III.

Step-by-Step Solution

Answer

Stage I: -16920.42 N, Stage II: -3117.79 N, Stage III: -29501.09 N.

Step 1: Convert Speeds to m/s

First, we need to convert the speeds from km/h to m/s because SI units will give us more precision in calculations.Initial Speed for Stage I: \[ 19300 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 5361.11 \text{ m/s} \]Final Speed for Stage I which is also Initial Speed for Stage II: \[ 1600 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 444.44 \text{ m/s} \]Final Speed for Stage II which is also Initial Speed for Stage III:\[ 321 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 89.17 \text{ m/s} \]Final Speed for Stage III:\[ 0 \text{ m/s} \]

Step 2: Calculate Accelerations for Each Stage

Using the formula for constant acceleration \[ a = \frac{v_f - v_i}{t} \]we compute the acceleration for each stage.Stage I: \[ a_1 = \frac{444.44 \text{ m/s} - 5361.11 \text{ m/s}}{240 \text{ s}} = -20.46 \text{ m/s}^2 \]Stage II:\[ a_2 = \frac{89.17 \text{ m/s} - 444.44 \text{ m/s}}{94 \text{ s}} = -3.77 \text{ m/s}^2 \]Stage III:\[ a_3 = \frac{0 \text{ m/s} - 89.17 \text{ m/s}}{2.5 \text{ s}} = -35.67 \text{ m/s}^2 \]

Step 3: Calculate Forces on the Spacecraft

The force exerted on the spacecraft can be calculated using Newton's second law, \[ F = ma \]where \( m \) is the mass of the spacecraft, 827 kg.Force during Stage I:\[ F_1 = 827 \text{ kg} \times (-20.46 \text{ m/s}^2) = -16920.42 \text{ N} \]Force during Stage II:\[ F_2 = 827 \text{ kg} \times (-3.77 \text{ m/s}^2) = -3117.79 \text{ N} \]Force during Stage III:\[ F_3 = 827 \text{ kg} \times (-35.67 \text{ m/s}^2) = -29501.09 \text{ N} \]

Key Concepts

Mars gravity

Mars, often called the Red Planet, has a significantly different gravitational pull compared to Earth. On Mars, the acceleration due to gravity is about 0.379 times that of Earth's gravity. This means that if you weigh 100 kg on Earth, you would weigh only 37.9 kg on Mars.

This difference is essential for planning missions like the Mars Exploration Rover landings. Engineers must consider Mars' gravitational influence when designing spacecraft descent strategies and controlling landing speeds. This reduced gravity affects how forces, such as atmospheric friction and parachute deployment, interact with the spacecraft. While Earth gravity is 9.81 m/s extsuperscript{2}, Mars gravity checks in at approximately 3.72 m/s extsuperscript{2}. This gives engineers and scientists a unique challenge when predicting vehicle behavior during various stages of the landing process.

Understanding these variations in gravitational force is crucial for establishing stable landings and ensuring the success of mission objectives, such as collecting samples and testing materials.

Newton's second law

Newton's second law of motion is a fundamental concept in dynamics and plays a critical role in understanding how objects move. The law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. Mathematically, this is expressed as \[ F = ma \]where

• \( F \) is the force applied,
• \( m \) is the mass, and
• \( a \) is the acceleration.

In the context of the Mars landing exercise, this law helps determine the force exerted on the rover at different stages of its descent. Since the mass of the spacecraft remains constant (827 kg in this case), any change in its speed (acceleration or deceleration) results in a corresponding force.

During the landing sequence, the force exerted by atmospheric friction, parachutes, and retrorockets is crucial. They are responsible for decelerating the rover from high speeds to a safe landing velocity. Each mode decelerates the rover through its unique force, calculated by multiplying the craft's mass by the calculated acceleration for the stage. Knowing these forces helps engineers design and plan for successful landings.

Free-body diagram

A free-body diagram is a simple graphical illustration used to visualize the forces acting upon an object. In dynamics, it is crucial for analyzing motion, and it's particularly useful in the context of studying the Mars Exploration Rover's descent.

In each stage of the rover's landing:

• Stage I involved atmospheric friction as a force opposing the rover's initial high speed.
• Stage II used a parachute, applying an upward force to continue deceleration.
• Stage III utilized retrorockets for a final braking force.

To create an effective free-body diagram: - Identify the object. Here, it's the rover. - Indicate all forces acting on it with arrows. Arrows should represent force direction and relative magnitude. - Label each force clearly, such as gravitational force, atmospheric drag, parachute force, or rocket thrust.

These diagrams provide a visual way to balance forces and check that calculations comply with Newton’s laws. Engineers use them to predict how vehicles will behave under various forces in unfamiliar environments like Mars.

Kinematics

Kinematics is the branch of mechanics dealing with the motion of objects without considering the forces that cause the motion. In the Mars landing problem, kinematics helps break down the descent into stages defined by velocity changes over time.

Kinematic equations assist in determining distances traveled and acceleration rates during each stage. For instance, you use the equation \( v_f = v_i + at \), allowing you to find acceleration when the initial and final velocities and time are known.

In the Mars rover scenario,

• Stage I involved friction-inducing deceleration over time results in a new speed.
• Stage II calculated parachute effectiveness based on achieved new speed in 94 seconds.
• Stage III relied on retrorockets to bring velocity to zero efficiently and predictably.
  

By understanding these velocity transitions, one can estimate how long each stage should last, ensuring safe touchdown. Furthermore, analysis of kinematic data is crucial for mission planning, avoiding hazards, and designing spacecraft suited for Mars' unique environment.