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Chapter 4: Problem 89

A rocket blasts off from rest and attains a speed of \(45 \mathrm{~m} / \mathrm{s}\) in \(15 \mathrm{~s}\). An astronaut has a mass of \(57 \mathrm{~kg}\). What is the astronaut's apparent weight during takeoff?

Short Answer

Expert verified
The astronaut's apparent weight during takeoff is 729.6 N.

Step by step solution

01

Define Apparent Weight

The apparent weight of the astronaut is the normal force acting on the astronaut. During takeoff, this is the sum of the gravitational force and the force due to the rocket's acceleration.
02

Calculate Gravitational Force

The gravitational force (weight) acting on the astronaut can be calculated using the formula: \[ F_g = m imes g \] where \( m = 57 \text{ kg} \) is the mass of the astronaut and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity. Plugging in the values: \[ F_g = 57 imes 9.8 = 558.6 \text{ N} \].
03

Calculate Rocket's Acceleration

The acceleration of the rocket can be calculated using the formula: \[ a = \frac{v}{t} \] where \( v = 45 \text{ m/s} \) is the final velocity and \( t = 15 \text{ s} \) is the time taken. Plugging in the values: \[ a = \frac{45}{15} = 3 \text{ m/s}^2 \].
04

Calculate Force Due to Rocket's Acceleration

The force experienced by the astronaut due to the rocket's acceleration is given by: \[ F_a = m imes a \] where \( m = 57 \text{ kg} \) is the mass of the astronaut and \( a = 3 \text{ m/s}^2 \) is the acceleration. Plugging in the values: \[ F_a = 57 \times 3 = 171 \text{ N} \].
05

Calculate Apparent Weight

The apparent weight of the astronaut is the sum of the gravitational force and the force due to the rocket's acceleration: \[ F_{apparent} = F_g + F_a \] \[ F_{apparent} = 558.6 + 171 = 729.6 \text{ N} \].

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Key Concepts

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

Gravitational Force
Gravitational force is a fundamental concept that affects all objects with mass. It is the force that attracts two bodies towards each other. In the context of our exercise, the Earth exerts a gravitational force on the astronaut.

To find the gravitational force, we use the equation:
  • Formula: \( F_g = m \times g \)
  • \( m \) is the mass of the object (57 kg for the astronaut)
  • \( g \) is the acceleration due to Earth's gravity, approximately 9.8 m/s虏
By substituting these values, we compute the gravitational force: \[ F_g = 57 \times 9.8 = 558.6 \text{ N} \]

This value represents the weight of the astronaut when standing still at the Earth's surface.
Rocket Acceleration
Rocket acceleration is essential for understanding changes in motion during liftoff. It describes how a rocket鈥檚 velocity changes with time as it rises off the Earth's surface.

We calculate this acceleration using the formula:
  • Formula: \( a = \frac{v}{t} \)
  • \( v \) is the final velocity reached (45 m/s)
  • \( t \) is the time taken to reach this velocity (15 seconds)
Plug these values into the formula: \[ a = \frac{45}{15} = 3 \text{ m/s}^2 \]

This tells us how fast the speed of the rocket increases per second. Rocket acceleration creates additional force on the astronaut beyond what's felt from gravity.
Normal Force
Normal force is the support force exerted upon an object in contact with another stable object. For an astronaut in a rising rocket, this force is felt as a reaction to the rocket's acceleration.

It's crucial for calculating the apparent weight of the astronaut, as it includes both the gravitational force and the force due to the acceleration of the rocket.

Given:
  • Gravitational Force: 558.6 N
  • Force from Rocket Acceleration: 171 N
The apparent weight, experienced as normal force, is: \[ F_{apparent} = F_g + F_a \] \[ F_{apparent} = 558.6 + 171 = 729.6 \text{ N} \]

This calculated force is what the astronaut feels as their weight inside the accelerating rocket.
Newton's Laws of Motion
Newton's Laws of Motion are critical in understanding how forces affect the motion of objects. The concepts behind these laws help us analyze situations like a rocket's takeoff.

**First Law (Law of Inertia):**
  • An object at rest stays at rest, and an object in motion stays in motion unless acted upon by an external force.
  • The astronaut inside the rocket would remain at rest unless the rocket provides a force to propel them upwards.


**Second Law (Law of Acceleration):**
  • The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Represented by \( F = m \times a \).
  • It shows us that the acceleration of the rocket, and thus the force felt by the astronaut, changes directly based on the forces applied.


**Third Law (Action-Reaction Law):**
  • For every action, there is an equal and opposite reaction.
  • The rocket pushes downwards upon the Earth's surface; in reaction, the Earth pushes the rocket upwards, causing the rise and subsequent forces experienced by the astronaut.


Understanding these laws aids in grasping why the apparent weight changes during a rocket's launch.

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