An astronaut is inside a $2.25 × 10^6$ 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 force, in terms of the astronaut's weight $w$, does the rocket exert on her? Start with a free-body diagram of the astronaut.

Verified step by step guidance

1. Start by drawing a free-body diagram for the astronaut. The forces acting on her are: (1) the upward normal force exerted by the rocket (F_rocket), and (2) the downward gravitational force (F_gravity = w, where w is the astronaut's weight).
2. Apply Newton's second law of motion: \( F_{net} = ma \), where \( F_{net} \) is the net force acting on the astronaut, \( m \) is her mass, and \( a \) is her acceleration. The net force is given by \( F_{net} = F_{rocket} - F_{gravity} \).
3. Substitute \( F_{gravity} = w \) and \( a = 4g \) (since the maximum safe acceleration is 4 times the acceleration due to gravity, \( g \)). This gives \( F_{rocket} - w = m(4g) \).
4. Express the astronaut's mass \( m \) in terms of her weight \( w \): \( m = \frac{w}{g} \). Substitute this into the equation: \( F_{rocket} - w = \frac{w}{g}(4g) \).
5. Simplify the equation to solve for \( F_{rocket} \): \( F_{rocket} = w + 4w = 5w \). Thus, the rocket exerts a force equal to 5 times the astronaut's weight on her.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. In this context, it helps identify the forces acting on the astronaut, including gravitational force and the thrust force from the rocket. By analyzing these forces, one can determine the net force and the resulting acceleration experienced by the astronaut.

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 crucial for calculating the force exerted by the rocket on the astronaut, as it allows us to relate the astronaut's weight and the required acceleration to reach the speed of sound without exceeding safe limits.

Acceleration due to Gravity

Acceleration due to gravity (g) is the acceleration experienced by an object due to Earth's gravitational pull, approximately 9.81 m/s². In this scenario, the astronaut's weight (w) is the force due to gravity acting on her mass. Understanding this concept is essential for determining the total force exerted by the rocket, as it must counteract both the gravitational force and provide additional force for the desired acceleration.