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Chapter 5: Problem 84

You are part of a design team for future exploration of the planet Mars, where \(g=3.7 \mathrm{m} / \mathrm{s}^{2} .\) An explorer is to step out of a survey vehicle traveling horizontally at 33 \(\mathrm{m} / \mathrm{s}\) when it is 1200 \(\mathrm{m}\) above the surface and then fall freely for 20 \(\mathrm{s}\) . At that time, a portable advanced propulsion system (PAPS) is to exert a constant force that will decrease the explorer's speed to zero at the instant she touches the surface. The total mass (explorer, suit, equipment, and PAPS) is 150 \(\mathrm{kg}\) . Assume the change in mass of the PAPS to be negligible. Find the horizontal and vertical components of the force the PAPS must exer, and for what interval of time the PAPS must exert it. You can ignore air resistance.

Short Answer

Expert verified
Vertical force is -1447.5 N, horizontal force is 0 N, exertion time is 12.44 s.

Step by step solution

01

Determine Initial Vertical Velocity

The explorer steps out of the vehicle and is initially moving horizontally with a velocity of 33 \( \mathrm{m/s} \). Therefore, the initial vertical velocity \( v_{0y} \) is 0 \( \mathrm{m/s} \).
02

Calculate Vertical Displacement at 20 Seconds

Use the formula for vertical displacement \( y = v_{0y}t + \frac{1}{2}gt^2 \) to find the vertical displacement after 20 seconds. Given \( g = 3.7 \mathrm{m/s^2} \), \( y = 0 + \frac{1}{2}(3.7)(20)^2 = 740 \mathrm{m} \).
03

Calculate Remaining Distance to Surface

The total height is \( 1200 \mathrm{m} \), so the distance remaining to fall after 20 seconds is \( 1200 - 740 = 460 \mathrm{m} \).
04

Determine Final Vertical Velocity at 20 Seconds

Use the formula \( v_{y} = v_{0y} + gt \) to find the final velocity after 20 seconds. \( v_{y} = 0 + (3.7)(20) = 74 \mathrm{m/s} \).
05

Calculate Deceleration Required to Stop the Explorer

To reduce the velocity from \( 74 \mathrm{m/s} \) to \( 0 \mathrm{m/s} \) over a vertical distance of 460 m, use the equation \( v^2 = u^2 + 2as \). Rearranging gives \( a = \frac{-u^2}{2s} = \frac{-(74)^2}{2 \times 460} = -5.95 \mathrm{m/s^2} \), where \( u = 74 \mathrm{m/s} \) and \( v = 0 \mathrm{m/s} \).
06

Calculate Force Exerted by PAPS Vertically

The vertical force required by PAPS is found using \( F_y = m(a - g) \). Substituting the values, we get \( F_y = 150(-5.95 - 3.7) = -1447.5 \mathrm{N} \). The negative sign indicates that the force is in the upward direction.
07

Find Horizontal Velocity Component

The horizontal velocity remains constant at \( 33 \mathrm{m/s} \) since there is no horizontal force acting.
08

Calculate Horizontal Force Exerted by PAPS

Since the horizontal speed doesn't change, the horizontal component of the force is zero: \( F_x = 0 \mathrm{N} \).
09

Calculate Time Interval for PAPS Exertion

Use the equation \( v = u + at \) where \( v = 0 \), \( u = 74 \), and \( a = -5.95 \). Solving for \( t \), \( 0 = 74 + (-5.95)t \), so \( t \approx 12.44 \mathrm{s} \).

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Key Concepts

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

Mars gravity
Mars, known as the Red Planet, has a significantly lower gravitational force compared to Earth. On Mars, the gravitational acceleration is only 3.7 m/s虏. This means anything that falls on Mars, including astronauts and equipment, will experience this reduced gravity.
This lower gravity impacts how objects move, making them fall more slowly than on Earth, where gravity is approximately 9.8 m/s虏. This weaker gravitational pull allows explorers more time to react when moving close to the planet's surface.
In our exercise, understanding Mars' gravity is crucial because it affects how fast our explorer will fall and how much force is needed to stop or slow down their descent. When the explorer jumps off the vehicle, they initially fall with no vertical speed, as only Mars' gravitational pull acts on them. This detail is crucial for calculating the forces and times involved in the mission.
Vertical and horizontal forces
When analyzing the motion of the explorer, it's important to consider both vertical and horizontal forces.
Initially, the explorer moves horizontally at the same speed as the vehicle, which is 33 m/s. This horizontal motion continues at a constant speed since no additional horizontal forces are acting on the explorer. So, the horizontal force component remains zero.
  • **Vertical Force:** Initially, the explorer is subject to Mars' gravity, accelerating downwards at 3.7 m/s虏, which determines the speed increase as they fall.
  • **Deceleration Force:** To safely land the explorer, the PAPS must apply an upward force to decelerate them to a stop when reaching the surface. Calculating this force involves determining the deceleration required and applying Newton's second law to find the necessary exertion from the propulsion system.
Understanding these forces helps determine how the propulsion system needs to act in the vertical direction while the horizontal motion goes unaltered.
PAPS propulsion system
The Portable Advanced Propulsion System (PAPS) is a critical tool for the explorer's safe descent on Mars. This system is designed to exert controlled forces to manipulate the explorer's speed.
  • **Vertical Force Exertion:** Once activated, the PAPS must generate an upward thrust powerful enough to counter the current fall's speed, effectively bringing it to zero just as the explorer reaches the Martian surface. This requires precise calculation of both the force magnitude and duration.
  • **Operation Duration:** In our scenario, once the explorer has fallen for approximately 20 seconds, the PAPS must engage for about 12.44 seconds to decelerate the fall fully. This involves calculating how long the force should be applied using kinematic principles.
The PAPS allows explorers to fine-tune their descent, ensuring a safe and controlled landing, which is vital for human exploration missions on Mars.
Kinematics equations
Kinematics deals with the motion of objects without considering the forces causing the motion. It employs equations to describe various aspects of motion, such as displacement, velocity, and acceleration over time.
In this Mars exploration scenario, the kinematics equations are pivotal in determining various factors:
  • **Vertical Displacement and Time:** Using the equation for vertical displacement ,i.e., \(y = v_{0y}t + \frac{1}{2}gt^2\), we can calculate how far the explorer falls within a certain period.
  • **Final Velocity:** The kinematics equation used here, \(v = u + at\), helps predict the explorer's speed after a given time, influencing the PAPS's operation duration and the force it needs to exert.
  • **Deceleration Calculation:** This involves using \(v^2 = u^2 + 2as\), allowing us to determine the deceleration required to stop the explorer.
These equations form the backbone of planning the explorer's safe descent by providing the necessary calculations for both distance and velocity adjustments.

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Most popular questions from this chapter

A box with mass \(m\) is dragged across a level fioor having a coefficient of kinctic friction \(\mu_{k}\) by a rope that is pulled upward at an angle \(\theta\) above the horizontal with a force of magnitude \(F .(\text { a) In }\) terms of \(m, \mu_{k}, \theta,\) and \(g,\) obtain an expression for the magnitude of force required to move the box with constant speed. (b) Knowing that you are studying physics, a CPR instructor asks you how much force it would take to slide a \(90-\mathrm{kg}\) patient across a floor at constant speed by pulling on him at an angle of \(25^{\circ}\) above the borizontal. By dragging some weights wrapped in an old pair of pants down the hall with a spring balance, you find that \(\mu_{k}=0.35 .\) Use the result of part (a) to answer the instructor's question.

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