Problem 11 - University Physics with Modern Physics

Question

An astronaut is inside a \(2.25 \times 10^{6} \mathrm{kg}\) rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound \((331 \mathrm{m} / \mathrm{s})\) as quickly as possible, but you also do not want the astronaut to black out. Medical tests have shown that astronauts are in danger of blacking out at an acceleration greater than 4\(g .\) (a) What is the maximum thrust the engines of the rocket can have to just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of her weight \(w,\) does the rocket exert on the astronatt? 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?

Step-by-Step Solution

Answer

(a) Maximum thrust is \(1.102125 \times 10^8 \text{N}\). (b) Force on astronaut is \(5w\). (c) Shortest time is \(8.44 \text{s}\).

Step 1: Define Variables and Constants

For all calculations, define:- Gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \).- Mass of the rocket, \( m = 2.25 \times 10^6 \, \text{kg} \).- Speed of sound, \( v_s = 331 \, \text{m/s} \).- Maximum safe acceleration for the astronaut, \( a_{max} = 4g = 4 \times 9.81 \, \text{m/s}^2 \).

Step 2: Newton's Second Law for Rocket

To find the maximum thrust, use Newton's second law:\[F_{net} = m \cdot a = T - mg\]where \( T \) is thrust, \( m \cdot g \) is gravitational force. Substitute \( a = 4g \):\[m \cdot 4g = T - mg \, \Rightarrow \, T = m \cdot 5g\]

Step 3: Calculate Maximum Thrust

Substitute the known values:\[T = 2.25 \times 10^6 \, \text{kg} \times 5 \times 9.81 \, \text{m/s}^2\]Calculate:\[T = 1.102125 \times 10^8 \, \text{N}\]

Step 4: Free-Body Diagram for Astronaut

Consider the rocket exerting a force on the astronaut equivalent to the astronaut's weight plus the effective force from acceleration. If \( w = m_a \cdot g \):\[F_{exerted} = m_a \cdot 5g = 5w\]

Step 5: Determine Shortest Time to Reach Speed of Sound

From physics, \( v = a \cdot t \). Solve for \( t \) using \( a = 4g \):\[t = \frac{v_s}{4g} = \frac{331}{4 \times 9.81}\]Calculate:\[t \approx 8.44 \, \text{s}\]

Step 6: Compile Results

Summarize the answers:(a) Maximum thrust without blacking out: \( 1.102125 \times 10^8 \, \text{N} \).(b) Force on astronaut as a multiple of weight: \( 5w \).(c) Shortest time to reach speed of sound: \( \approx 8.44 \, \text{s}\).

Key Concepts

Acceleration Limits

Acceleration limits are crucial when it comes to human spaceflight. Astronauts experience significant forces during launch, expressed as multiples of gravity, or "g-forces." In this case, medical research shows that astronauts risk blacking out at accelerations above four times gravity, or 4g.

Understanding these limits helps ensure astronaut safety during rapid acceleration. The human body can tolerate some significant acceleration, but above certain thresholds, the risk of loss of consciousness increases due to reduced blood flow to the brain. Reducing acceleration within safe limits is essential during a rocket's launch phase.

Thus, engineers need to design the rocket engines to produce enough thrust to achieve their mission goals without exceeding these critical limits.

Newton's Second Law

Newton's second law of motion is a fundamental concept used to understand how forces affect motion. It states that the force on an object is equal to the mass of the object multiplied by its acceleration, expressed as \( F = m \cdot a \).

In the context of a rocket, this law can calculate the net force required to produce a certain acceleration. For a rocket blasting off, the net force must not only overcome gravity but also provide the additional acceleration needed for liftoff.

To ensure the astronaut remains safe, the rocket's thrust must be precisely calculated to remain within safe acceleration limits while still providing the needed force for ascent. This ensures that the rocket achieves its desired velocity while keeping g-forces within safe boundaries for its human occupants.

Free-Body Diagram

A free-body diagram represents the forces acting on a single object, helping visualize and solve physics problems. In this rocket scenario, free-body diagrams for both the rocket and the astronaut are crucial.

For the rocket, the diagram shows two main forces: the upward thrust from the engines and the downward gravitational force, \( mg \). Calculating the net force involves subtracting gravity from thrust.

For the astronaut, the diagram helps us understand how the rocket's force translates into the astronaut's experience. Here, the force experienced is the combination of gravitational force, \( m_a \cdot g \), and the additional force due to acceleration, resulting in a total of \( 5w \) when \( w \) is their weight in normal conditions.

Maximum Thrust Calculation

Calculating maximum thrust involves ensuring that the rocket's force remains within safe acceleration limits while achieving mission objectives. This calculation uses the formula derived from Newton's second law.

Given that the astronaut's safe acceleration is 4g, we include gravitational force \( mg \) as part of the total required force. This yields the formula for maximum thrust, \( T = m \cdot 5g \). This approach ensures the rocket can accelerate to the speed of sound without exceeding safe limits.

By substituting known values like rocket mass and gravitational acceleration, the calculation gives an engine thrust of approximately \( 1.102125 \times 10^8 \, \text{N} \), which is sufficient for reaching desired speeds safely and efficiently.

Speed of Sound

The speed of sound is significant because it's a milestone for aerospace vehicles that indicates rapid velocity. It's approximately 331 m/s at sea level in air.

Reaching the speed of sound as quickly as possible is often a target in space missions, particularly during liftoff. However, reaching it must be done within acceleration limits to avoid endangering the crew's health.

Using the rocket's designed acceleration, to reach 331 m/s safely, the law \( v = a \cdot t \) helps determine the shortest time, mathematically calculated to be around 8.44 seconds, emphasizing efficient engineering and mission planning.