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Chapter 4: Problem 106

Medical, Astro The space shuttle takes off vertically from rest with an acceleration of \(29 \mathrm{~m} / \mathrm{s}^{2}\). What force is exerted on a \(75-\mathrm{kg}\) astronaut during takeoff? Express your answer in newtons and also as a multiple of the astronaut's weight on Earth.

Short Answer

Expert verified
The force exerted on the astronaut during takeoff is 2175 newtons and is approximately three times the astronaut's weight on Earth.

Step by step solution

01

Determine the Force Exerted on the Astronaut

Using Newton's second law of motion, the force exerted on the astronaut during takeoff can be calculated as the mass of the astronaut times the acceleration of takeoff. F = ma = (75 kg) * (29 m/s虏) = 2175 newtons.
02

Determine the Astronaut's Weight on Earth

The weight of an astronaut (or any object) on Earth can be calculated as the mass of the astronaut times the force of gravity. W = mg = (75 kg) * (9.8 m/s虏) = 735 newtons.
03

Express the Force as a Multiple of the Astronaut's Weight

By dividing the force exerted on the astronaut during takeoff (from Step 1) by the astronaut's weight on Earth (from Step 2), we obtain the force as a multiple of the astronaut's weight. Multiple = F/W = 2175 newtons / 735 newtons = 2.96
04

Interpret the Result

The result of 2.96 implies that the force exerted on the astronaut during takeoff is approximately three times the astronaut's weight on Earth.

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Key Concepts

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

Force Exertion
In the context of the space shuttle's takeoff, understanding how force is exerted is pivotal. Our guiding principle here is Newton's Second Law of Motion. This law succinctly states that force is the product of mass and acceleration. To visualize this, imagine the space shuttle racing skyward. The acceleration it applies to overcome Earth's gravity and propel itself into space is tremendous.

In our example, we have an astronaut with a mass of 75 kg who experiences a takeoff acceleration of 29 m/s虏. Applying Newton's law, the force exerted on the astronaut is calculated as follows:
  • The formula used is: \( F = ma \)
  • By substituting the values, \( F = 75 \, \text{kg} \times 29 \, \text{m/s}^2 \)
  • The result is a force of 2175 newtons. This immense force is crucial for the shuttle's upward journey and the astronaut's safety.
Acceleration Due to Gravity
Gravity's role is an unwavering constant in our daily lives and crucial for understanding weight. The Earth's gravity gives an acceleration of approximately 9.8 m/s虏 to falling objects. This figure is somewhat universal, providing a predictable behavior to any object subject to Earth's pull.

In everyday examples, it influences how fast a ball falls when you drop it or how quickly you descend after jumping. For our astronaut, this acceleration defines their weight on Earth. This natural acceleration is a foundation concept when calculating how much the same astronaut would weigh under different forces, such as during a space shuttle launch.
Weight Calculation
Weight is a concept often confused with mass, yet they are quite different. While mass refers to the amount of matter in an object, weight is the force that gravity exerts on that object. When we speak of an object's weight on Earth, we're referring to the gravitational force acting on it.

To determine the weight of the 75 kg astronaut on Earth, we use:
  • The weight formula: \( W = mg \)
  • Here, \( m = 75 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \)
  • This calculation yields a weight of 735 newtons. Knowing the astronaut鈥檚 weight is a baseline for calculating the forces involved during more extreme conditions, like a shuttle takeoff.
Multiple of Weight
Expressing force as a multiple of weight provides a helpful comparison, especially in contexts of varying gravitational forces like space travel. In our shuttle scenario, the force exerted on the astronaut during takeoff can be expressed in terms of his Earth weight.
  • We have already calculated the force exerted during takeoff as 2175 newtons.
  • And the astronaut's Earth weight is 735 newtons.
  • To find the multiple, we use: \( \text{Multiple} = \frac{F}{W} = \frac{2175}{735} \approx 2.96 \)
This means that the force experienced by the astronaut is about three times stronger than his weight on the ground. Such multiples are crucial for designing equipment that can safely withstand the forces involved in space travel.

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