Problem 15 - College Physics

Question

Stunt pilots and fighter pilots who fly at high speeds in a downward-curving arc may experience a "red out," in which blood is forced upward into the flier's head, potentially swelling or breaking capillaries in the eyes and leading to a reddening of vision and even loss of consciousness. This effect can occur at centripetal accelerations of about \(2.5 g\) 's. For a stunt plane flying at a speed of \(320 \mathrm{~km} / \mathrm{h},\) what is the minimum radius of downward curve a pilot can achieve without experiencing a red out at the top of the arc? (Hint: Remember that gravity provides part of the centripetal acceleration at the top of the arc; it's the acceleration required in excess of gravity that causes this problem.)

Step-by-Step Solution

Answer

The minimum radius is approximately 322.17 meters.

Step 1: Understand the Problem

The problem involves calculating the minimum radius for a plane's downward curve to avoid exceeding a centripetal acceleration of \(2.5 g\). The acceleration of gravity (\(g\)) contributes to this centripetal acceleration.

Step 2: Convert Speed Units

Convert the plane's speed from km/h to m/s. The speed is \(320\, \text{km/h}\). To convert, use the relation: \(1\, \text{km/h} = \frac{1000}{3600}\, \text{m/s}\). Therefore, \(320\, \text{km/h} = \frac{320 \times 1000}{3600}\, \text{m/s} \approx 88.89\, \text{m/s}\).

Step 3: Apply Formula for Centripetal Force

The formula for centripetal acceleration is \(a_c = \frac{v^2}{r}\), where \(v\) is the velocity and \(r\) is the radius of the curve. Given the total centripetal acceleration is \(2.5g\), where \(g = 9.81\, \text{m/s}^2\), we set the equation to \(a_c = 2.5 \times 9.81\, \text{m/s}^2\).

Step 4: Solve for the Minimum Radius

Rearrange the centripetal acceleration formula to solve for \(r\): \(r = \frac{v^2}{a_c}\). Substituting \(v = 88.89\, \text{m/s}\) and \(a_c = 2.5 \times 9.81\, \text{m/s}^2 = 24.525\, \text{m/s}^2\), we get: \[ r = \frac{(88.89)^2}{24.525} \approx 322.17 \text{ meters} \].

Key Concepts

Red Out

Flying at high speeds in a downward arc can cause an effect known as "red out". This happens when gravity, along with the centrifugal forces, pushes blood upwards into the head.
This force can cause blood vessels in the eyes to swell or even break. The result is a reddish tint in vision or even temporary loss of consciousness.
When the centripetal acceleration exceeds approximately 2.5 times that of regular gravity (\(2.5g\)), the risk of a red out greatly increases. This situation is dangerous as it can impair pilots just when they need to be most alert.
Therefore, understanding the forces and calculations involved is crucial for stunt pilots to ensure their maneuvers remain safe.

Physics Problem Solving

Solving physics problems often begins with a clear understanding of the scenario. Breaking down the problem into smaller, manageable pieces can help.
Here is a practical approach:

• Understanding the Problem: First, identify what is being asked, such as calculating the minimum radius in this scenario.
• Converting Units: Physics problems frequently require converting units for consistency. Here, converting speed from km/h to m/s is essential.
• Applying Equations: Knowing which physical equations to use is key, such as the formula for centripetal acceleration.
• Solution Execution: Employ the formulas and solve step-by-step to find the needed variable, like the radius here.

This structured approach can simplify complex physical scenarios and make problem-solving both manageable and effective.

Minimum Radius Calculation

To avoid the dangerous effects of a red out, it's essential to calculate the minimum radius of a downward curve. This prevents the plane's centripetal acceleration from exceeding safe limits.
Using the formula for centripetal acceleration, \(a_c = \frac{v^2}{r}\), allows us to solve for the radius \(r\). Given the velocity \(v\) and the acceleration limit \(a_c\) (which in our case is \(2.5g\)), you can rearrange this formula:\[r = \frac{v^2}{a_c} \] Substitute the values (with \(v = 88.89 \text{ m/s}\) and \(a_c = 24.525 \text{ m/s}^2\)) to find: \[ r \approx 322.17 \text{ meters} \] This shows how you can ensure you're navigating a safe path by accurately calculating the minimized radius to reduce the risk of a red out.

Stunt Pilots

Stunt pilots face extreme conditions when performing aerial maneuvers. These involve high speeds and tight turns. Understanding physics is crucial to staying safe in such demanding environments.
They rely heavily on calculations to determine safe limits and design stunts. Investigating factors like speed, the gravitational pull, and the radius of turns can dictate a successful flight path.

• Simulation: Pilots often use simulations to test scenarios before real flight.
• Training: Intensive training helps pilots respond quickly to unexpected forces.
• Equipment: Technology aids in monitoring and maintaining safe maneuvers during stunts.

Being informed about these elements ensures that stunt pilots can perform thrilling displays while remaining safe and composed in the skies.