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 $4g$. 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.

Verified step by step guidance

1. Start by identifying the forces acting on the rocket. The two main forces are the thrust force \( F_{\text{thrust}} \) acting upward and the gravitational force \( F_{\text{gravity}} \) acting downward. The net force \( F_{\text{net}} \) determines the rocket's acceleration.
2. Write the equation for the net force using Newton's second law: \( F_{\text{net}} = F_{\text{thrust}} - F_{\text{gravity}} = m a \), where \( m \) is the mass of the rocket and \( a \) is the acceleration.
3. The maximum acceleration the astronaut can safely experience is \( a = 4g \), where \( g = 9.8 \, \text{m/s}^2 \). Substitute \( a = 4g \) into the equation: \( F_{\text{thrust}} - F_{\text{gravity}} = m (4g) \).
4. Calculate the gravitational force \( F_{\text{gravity}} \) using \( F_{\text{gravity}} = m g \). Substitute \( F_{\text{gravity}} \) into the equation: \( F_{\text{thrust}} = m (4g) + m g \).
5. Simplify the expression for \( F_{\text{thrust}} \): \( F_{\text{thrust}} = m (4g + g) = m (5g) \). Substitute the given mass of the rocket \( m = 2.25 \times 10^6 \; \text{kg} \) and the value of \( g = 9.8 \; \text{m/s}^2 \) to find the maximum initial thrust.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. In the context of the rocket, it helps identify the forces such as thrust, gravitational force, and any other acting forces. By analyzing these forces, one can determine the net force and subsequently the acceleration of the rocket, which is crucial for understanding how to achieve the desired speed without exceeding safe acceleration limits.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, expressed as F = ma. This principle is essential for calculating the maximum thrust the rocket can generate while ensuring that the acceleration does not exceed 4g (where g is the acceleration due to gravity). Understanding this law allows for the determination of the thrust needed to achieve the desired acceleration safely.

Acceleration and G-Forces

Acceleration is the rate of change of velocity of an object, and it is often expressed in terms of g-forces, where 1g is equivalent to the acceleration due to Earth's gravity (approximately 9.81 m/s²). In this scenario, the astronaut can tolerate a maximum acceleration of 4g, which translates to an upper limit of about 39.24 m/s². This concept is critical for ensuring that the rocket's thrust does not induce excessive acceleration that could lead to blackouts for the astronauts.