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Chapter 5: Problem 11

An astronaut is inside a 2.25 \(\times\) 10\(^6\) kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4\(g\). (a) What is the maximum initial thrust this rocket's engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of the astronaut's weight \(w\), does the rocket exert on her? Start with a free-body diagram of the astronaut. (c) What is the shortest time it can take the rocket to reach the speed of sound?

Short Answer

Expert verified
(a) Maximum thrust is 1.1025 脳 10鈦� N. (b) Rocket exerts force 5w on the astronaut. (c) Shortest time to reach speed of sound is about 8.44 s.

Step by step solution

01

Identify Given Data

We begin by extracting the given data: the mass of the rocket \( m = 2.25 \times 10^6 \) kg, acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \), and the speed of sound \( v = 331 \, \text{m/s} \). The fear of blackouts occurs at accelerations greater than \( 4g \).
02

Calculate Maximum Safe Acceleration

The maximum acceleration \( a \) the astronaut can withstand is \( 4g \), which gives \( a = 4 \times 9.8 \, \text{m/s}^2 = 39.2 \, \text{m/s}^2 \).
03

Analyze Rocket's Forces (a)

Construct a free-body diagram for the rocket. The forces acting are gravitational force \( F_g = mg \) and the thrust \( T \). Using Newton's second law, we have \( T - mg = ma \). Hence, \( T = m(a + g) \).
04

Solve for Maximum Thrust (a)

Substitute the values into the equation for \( T \):\[ T = 2.25 \times 10^6 \, \text{kg} \times (39.2 + 9.8) \, \text{m/s}^2 = 2.25 \times 10^6 \, \text{kg} \times 49 \, \text{m/s}^2 \]This results in a thrust \( T = 1.1025 \times 10^8 \, \text{N} \).
05

Analyze Astronaut's Forces (b)

Create a free-body diagram for the astronaut. If the "weight" of the astronaut is \( w \), the force exerted by the rocket on the astronaut follows the relation \( F_{rocket} = m_{astronaut} \times (a + g) \). Since \( a = 4g \), this simplifies to \( F_{rocket} = 5w \).
06

Calculate Minimum Time (c)

To find the shortest time \( t \) needed to reach 331 m/s, use the kinematic equation \( v = at \). We found earlier \( a = 39.2 \) m/s虏, thus:\[ t = \frac{v}{a} = \frac{331 \, \text{m/s}}{39.2 \, \text{m/s}^2} \approx 8.44 \, \text{s} \].

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Key Concepts

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

Understanding Free-Body Diagrams
Free-body diagrams are essential to visualizing and analyzing forces acting on an object. They simplify complex physical interactions into straightforward representations. In these diagrams, objects are usually represented as simple shapes (like boxes), and arrows indicate the forces acting upon them.

For instance, in our rocket problem, the free-body diagram includes:
  • The Earth's gravitational force pulling the rocket downwards.
  • The thrust acting upwards produced by the rocket's engines to lift it off the ground.
The length and direction of the arrows are indicative of the forces' relative magnitudes and directions. To correctly apply Newton's laws, placing these forces in a diagram helps clarify opposing forces and their result on the object's motion.

Such clarity is crucial when determining critical factors like the rocket's needed thrust or its maximum acceleration limits.
The Acceleration Limit of the Rocket
Acceleration limits are vital to consider, especially when they can impact the safety of passengers. In rocket physics, acceleration is often expressed in terms of gravity (\( g \), where \( g = 9.8 \, \text{m/s}^2 \)).

An astronaut is at risk of blacking out when subjected to an acceleration greater than \( 4g \). This simply means the acceleration should not exceed \( 39.2 \, \text{m/s}^2 \), derived as \( 4 \times 9.8 \, \text{m/s}^2 \). This consideration is critical not just in theoretical calculations but also in practical applications where human safety and comfort are paramount.

This acceleration limit ensures a safe journey, avoiding situations that could lead to unconsciousness due to excessive forces on the astronaut's body.
Application of Newton's Second Law
Newton's second law forms the backbone of calculating forces and motion. It states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration: \( F = ma \).

For our problem, this law helps calculate the maximum thrust needed for the rocket to safely reach its acceleration limit without risking blackout. The thrust had to overcome not only the rocket's gravitational pull but also provide the necessary upward acceleration. Hence, the formula applied was \( T = m(a + g) \), ensuring that the sum of thrust minus gravity equals the resultant upward acceleration.

This manipulation of the second law allows us to precisely compute the thrust, ensuring both the ascent and safety of the crew.
Achieving the Speed of Sound
The speed of sound, which is approximately \( 331 \, \text{m/s} \) in air, marks a significant milestone in aerodynamics and rocket science. Reaching this speed efficiently demands precise calculation based on acceleration capabilities.

To determine the shortest time needed to achieve the speed of sound, a kinematic equation \( v = at \) is used. Given the maximum acceleration that ensures safety (\( a = 39.2 \, \text{m/s}^2 \)), the time \( t \) it takes to reach \( 331 \, \text{m/s} \) can be calculated as \( \frac{v}{a} \), resulting in approximately \( 8.44 \, \text{s} \).

These calculations provide not only an exciting glimpse into the capabilities of rocket technology but also underline the importance of balancing speed with safety. Achieving high speeds safely is central to advancements in space exploration and travel.

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Most popular questions from this chapter

One problem for humans living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates "artificial gravity" at the outside rim of the station. (a) If the diameter of the space station is 800 m, how many revolutions per minute are needed for the "artificial gravity" acceleration to be 9.80 m/s\(^2\)? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface (3.70 m/s\(^2\)). How many revolutions per minute are needed in this case?

A 5.00-kg box sits at rest at the bottom of a ramp that is 8.00 m long and is inclined at 30.0\(^\circ\) above the horizontal. The coefficient of kinetic friction is \(\mu_k\) = 0.40, and the coefficient of static friction is \(\mu_s\) = 0.43. What constant force \(F\), applied parallel to the surface of the ramp, is required to push the box to the top of the ramp in a time of 6.00 s?

You are called as an expert witness in a trial for a traffic violation. The facts are these: A driver slammed on his brakes and came to a stop with constant acceleration. Measurements of his tires and the skid marks on the pavement indicate that he locked his car's wheels, the car traveled 192 ft before stopping, and the coefficient of kinetic friction between the road and his tires was 0.750. He was charged with speeding in a 45-mi/h zone but pleads innocent. What is your conclusion: guilty or innocent? How fast was he going when he hit his brakes?

The Cosmo Clock 21 Ferris wheel in Yokohama, Japan, has a diameter of 100 m. Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s). (a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 N at the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if the passenger's apparent weight at the highest point were zero? (d) What then would be the passenger's apparent weight at the lowest point?

A 25.0-kg box of textbooks rests on a loading ramp that makes an angle \(\alpha\) with the horizontal. The coefficient of kinetic friction is 0.25, and the coefficient of static friction is 0.35. (a) As \(\alpha\) is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0 m along the loading ramp?
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