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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. How much time elapses between the first and second bounces? A. \(1.38 \mathrm{s}\) B. \(2.58 \mathrm{s}\) C. \(5.15 \mathrm{s}\) D. \(5.33 \mathrm{s}\)

Step by step solution

01

Break Down Initial Velocity

The rover rebounds with an initial velocity of 9.92 m/s at an angle of 75° above the horizontal. The horizontal and vertical components of this velocity can be calculated using trigonometric functions. The initial vertical velocity \(v_{i_y}\) is given by \(v_{i_y} = v_i \cdot \sin(\theta)\). So, \(v_{i_y} = 9.92 \cdot \sin(75^\circ)\).

02

Calculate Vertical Velocity Component

Using the equation from Step 1, calculate the vertical velocity component: \(v_{i_y} = 9.92 \cdot \sin(75^\circ) \approx 9.58 \mathrm{m/s}\).

03

Use Kinematic Equation for Time of Flight

The time between the first and second bounces involves the rover going up and coming back down to Martian surface level. This is a vertical motion under Martian gravity. Use the kinematic equation \(v = v_i + at\), where final vertical velocity \(v = 0\) at the highest point, \(a\) is the acceleration due to gravity \(-3.72\, \mathrm{m/s^2}\), and initial velocity \(v_{i}\) is from Step 2. The time \(t\) to reach the peak is \(t = \frac{v_i}{a} = \frac{9.58}{3.72}\).

04

Calculate Time to Reach Peak

Calculate the time to reach the peak using the equation: \(t = \frac{9.58}{3.72} \approx 2.575\, \mathrm{s}\).

05

Determine Total Time for One Bounce

Double the time calculated in Step 4 because the time going up equals the time coming down: \(2 \times 2.575 = 5.15 \mathrm{s}\).

06

Identify the Correct Answer

From the calculations, the time elapsed between the first and second bounces is \(5.15 \mathrm{s}\). Therefore, the correct answer is C.

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Key Concepts

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

Kinematic Equations

Kinematic equations are a set of mathematical formulas used to describe the motion of objects. They help in predicting the future position and velocity of moving bodies under uniform acceleration, which is a common scenario in physics problems. These equations include variables such as initial velocity, final velocity, acceleration, time, and displacement.
To solve problems like the Mars rover's motion, you typically use:

• The first equation: \( v = v_i + at \), which relates the final velocity \( v \), initial velocity \( v_i \), acceleration \( a \), and time \( t \).
• The second equation: \( s = v_i t + \frac{1}{2} a t^2 \), used to find the displacement \( s \) of an object.

These equations make it easier to break down complex motion problems into manageable parts by focusing on the essential parameters of the motion, such as speed and direction.

Martian Gravity

Gravity on Mars, often referred to as Martian gravity, is weaker than Earth's gravity. It is approximately \(3.72\, \mathrm{m/s^2}\) compared to Earth's \(9.81\, \mathrm{m/s^2}\). This difference impacts how objects move on Mars, affecting their speed and trajectory when they fall or when thrown, like the Mars rover making a bounce upon landing.
Understanding Martian gravity is crucial when calculating the time between bounces or the height the rover reaches after bouncing, as it directly influences acceleration in vertical motion.
The reduced gravity means that objects will fall more slowly and reach peak heights higher than they would on Earth, given the same conditions.

Projectile Motion

Projectile motion refers to a form of motion experienced by an object or projectile that is launched into the air and is influenced only by the force of gravity. In the case of the Mars rover, after a bounce, it follows a curved path determined by its velocity and angle of projection.
For any projectile, it is important to consider:

• The initial angle, which is 75° for our rover scenario.
• The initial speed, given as \(9.92 \mathrm{m/s}\).
• The influence of Martian gravity, which is \(3.72 \mathrm{m/s^2}\).

Combining these elements helps us break down the motion into its horizontal and vertical components for easier analysis. The horizontal motion remains constant since Martian gravity only affects the vertical component.

Vertical Motion

Vertical motion, a component of projectile motion, is specifically concerned with the upward and downward path of an object. For the Mars rover's trajectory, we deal with vertical motion till it bounces back off the Martian surface.
Key points in vertical motion calculations include:

• The initial vertical velocity, calculated using trigonometry as \(9.58 \mathrm{m/s}\).
• The acceleration due to Martian gravity, \(-3.72 \mathrm{m/s^2}\), acting downward.
• Using kinematic equations to determine the time intervals, such as using the formula \( t = \frac{v_i}{a} \) to find the time to the peak from the initial velocity.

Understanding vertical motion is essential for predicting how long it takes an object like the rover to rise to its peak and return to the surface.