91Ó°ÊÓ

When the twin Mars exploration rovers, Spirit and Opportunity, set down on the surface of the red planet in January of 2004 their method of landing was both unique and elaborate. After initial braking with retro rockets, the rovers began their long descent through the thin Martian atmosphere on a parachute until they reached an altitude of about \(16.7 \mathrm{m}\). At that point a system of four air bags with six lobes each were inflated, additional retro rocket blasts brought the craft to a virtual standstill, and the rovers detached from their parachutes. After a period of free fall to the surface, with an acceleration of \(3.72 \mathrm{m} / \mathrm{s}^{2}\), the rovers bounced about a dozen times before coming to rest. They then deflated their air bags, righted themselves, and began to explore the surface. Figure \(4-25\) shows a rover with its surrounding cushion of air bags making its first contact with the Martian surface. After a typical first bounce the upward velocity of a rover would be \(9.92 \mathrm{m} / \mathrm{s}\) at an angle of \(75.0^{\circ}\) above the horizontal. Assume this is the case for the problems that follow. What is the maximum height of a rover between its first and second bounces? A. \(2.58 \mathrm{m}\) B. \(4.68 \mathrm{m}\) C. \(12.3 \mathrm{m}\) D. \(148 \mathrm{m}\)

Step by step solution

01

Identify Vertical Velocity Component

The initial velocity is given as 9.92 m/s at an angle of 75.0° above the horizontal. To find the vertical component of this velocity, we use the sine function: \[ v_{y} = v \sin(\theta) = 9.92 \times \sin(75.0°) \].

02

Calculate Vertical Velocity Component

Calculate the vertical component using the sine of 75°:\[ v_{y} = 9.92 \times \sin(75.0°) \approx 9.58 \text{ m/s} \].

03

Use Kinematic Equation to Find Maximum Height

At the maximum height, the final vertical velocity is 0 m/s. Use the kinematic equation \[ v_{f}^2 = v_{i}^2 + 2a(y_{f} - y_{i}) \], where - \( v_{f} = 0 \text{ m/s}\) (final vertical velocity at max height),- \( v_{i} = 9.58 \text{ m/s}\) (initial vertical velocity),- \( a = -3.72 \text{ m/s}^2\) (since acceleration due to gravity on Mars is acting downwards),- \( y_{i} = 0 \text{ m}\) (initial height).Solve for \( y_{f} \), the maximum height.

04

Substitute Values and Solve for Maximum Height

Substituting into the kinematic equation:\[ 0 = (9.58)^2 + 2(-3.72)(y_{f} - 0) \].Simplify and solve for \( y_{f} \):\[ 0 = 91.6864 - 7.44y_{f} \]\[ 7.44y_{f} = 91.6864 \]\[ y_{f} = \frac{91.6864}{7.44} \approx 12.3 \text{ m} \].

05

Determine the Correct Answer from Given Options

The calculated maximum height is approximately 12.3 m, matching option C.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mars exploration

Mars exploration represents a giant leap in understanding the Red Planet and our solar system. In January 2004, NASA successfully landed twin rovers, Spirit and Opportunity, on Mars, paving the way for new discoveries. The landing method was a remarkable display of engineering creativity. It involved a sequence of deceleration techniques to safely reach the Martian surface.

• Firstly, retro rockets decelerated the descent after the initial entry.
• Parachutes deployed to control speed during their long descent through Mars's thin atmosphere.
• Airbags were then inflated to cushion the landing, coupled with additional retro rocket blasts to reach a near standstill before detachment from the parachutes.

This elaborate process highlights the technical challenges of Mars landings due to its unique surface and atmospheric conditions. The rovers were tasked with exploring Martian topography and geology, which contributed significantly to our understanding of Mars's history and its potential to support life.

Projectile motion

Projectile motion refers to the movement of an object thrown into the air, subject only to the force of gravity. In essence, a projectile experiences independent horizontal and vertical motion. This kind of motion can be perfectly examined using the example of the Mars rovers. When these rovers bounced on the surface of Mars, they followed a path dictated by the principles of projectile motion.

Components of Velocity

The motion of a projectile can be dissected into horizontal and vertical components. For instance, the upward velocity of a rover at 9.92 m/s at a 75° angle mainly consists of:

• Vertical Component: Determines how high the rover can ascend.
• Horizontal Component: Determines how far forward it travels horizontally.

These components are independent of each other, uniquely affecting the projectile's path. The vertical motion experiences constant gravitational acceleration, while the horizontal motion remains unaffected by gravity, providing a uniform motion. Solving problems involving projectile motion involves calculating these components individually and then analyzing the resultant trajectories.

Acceleration due to gravity

In the context of Mars exploration, understanding gravity's influence on the behavior of objects is crucial. Mars's gravity is significantly weaker than Earth's, at about 3.72 m/s². This weaker force affects how objects fall and move on Mars.

Impact on Free Fall and Bounces

The weaker gravitational pull means that objects, like the Mars rovers, fall slower compared to Earth. As the rovers made contact with the Martian surface, their bounce effect was influenced by this reduced acceleration due to gravity. This also affected how they gained altitude upon bouncing off the surface.

Calculating Motion with Mars's Gravity

When computing maximum heights or other motion specifics on Mars, one must consider this lower gravitational acceleration:

• Free-fall speed and duration.
• Height and distance calculations in projectile motion.
• The effects of initial velocity components in determining the path and peak of trajectories.

This altered gravity has a direct impact on the required calculations for the rover’s landings and movement performance on Mars.