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\(g^{\prime}\) 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 of the downward curve is approximately 536.67 meters.

Step 1: Convert Speed Units

First, convert the speed from kilometers per hour to meters per second. Use the conversion factor: \[ 1\,\text{km/h} = \frac{1}{3.6}\,\text{m/s} \] So, the speed \( v \) is calculated as: \[ v = 320\,\text{km/h} \times \frac{1}{3.6}\,\text{m/s} = 88.89\,\text{m/s} \]

Step 2: Apply Centripetal Force Equation

Utilize the formula for centripetal acceleration which is given by \[ a_c = \frac{v^2}{r} \]where \( a_c \) is the centripetal acceleration, \( v \) is the speed, and \( r \) is the radius. Rearrange this formula to solve for the minimum radius \( r \): \[ r = \frac{v^2}{a_c} \]

Step 3: Determine Resultant Centripetal Acceleration

At the top of the arc, the necessary centripetal acceleration is 2.5 \( g' \), where gravity already provides 1 \( g \). Thus, the additional acceleration needed is: \[ a_c = (2.5 - 1)g \]where \( g = 9.81\, \text{m/s}^2 \). Calculate \( a_c \): \[ a_c = 1.5 \times 9.81\, \text{m/s}^2 = 14.715\, \text{m/s}^2 \]

Step 4: Calculate Minimum Radius

Insert the calculated centripetal acceleration and speed into the formula for radius: \[ r = \frac{v^2}{a_c} = \frac{(88.89\,\text{m/s})^2}{14.715\,\text{m/s}^2} \] \[ r = \frac{7900.1121}{14.715} \] \[ r \approx 536.67\,\text{m} \]

Key Concepts

Kinetic Energy

Kinetic energy is the energy a body possesses due to its motion. It's what gives moving objects their ability to do work and engage in dynamic actions. The formula for calculating kinetic energy (KE) is given by:

• KE = \( \frac{1}{2} mv^2 \)
• Where \( m \) is the mass and \( v \) is the velocity of the object.

In the context of the stunt plane, kinetic energy plays a vital role as it is directly related to the speed of the plane. The higher the plane's speed, the greater its kinetic energy. This energy must be managed, especially when making precise maneuvers in the air.
While the problem primarily focuses on centripetal force and acceleration, understanding kinetic energy can help one appreciate how speed impacts the forces acting upon the pilot and plane. If too much kinetic energy is present without proper handling techniques, it can lead to dangerous situations like the "red out" phenomenon.

Acceleration

Acceleration is the rate of change of velocity of an object. In circular motion, particularly in the context of the stunt plane, it refers to how quickly the plane’s speed or direction is changing as it travels in a curve. Acceleration can be the result of speeding up, slowing down, or changing direction.

• In circular motion, acceleration is directed towards the center of the circle. This is known as centripetal acceleration.
• For the stunt plane, centripetal acceleration is crucial as it keeps the plane moving in a circular path, especially during arcs.
• The challenge is ensuring that the centripetal acceleration does not exceed what the pilot can comfortably handle. A balance is needed between maintaining speed and minimizing discomfort or danger.

Understanding how acceleration works in circular motion helps in comprehending why certain maneuvers can cause effects like the "red out". Properly calculating the acceleration ensures the pilot remains safe while performing complex aerial tricks.

Circular Motion

Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It involves constant change in direction, necessitating an inward force to maintain the motion, known as centripetal force.

• This inward force is what keeps the stunt plane traveling in a circular arc rather than flying outwards.
• The centripetal force required is provided by a combination of the plane's thrust, gravity, and aerodynamic forces depending on its position in the arc.
• At the top of a downward arc, gravity contributes a significant portion of this force. The plane must generate additional acceleration to maintain its path.
• This contributes to the difficulty of maintaining manageable forces on the pilot and ensuring safe flight paths.

Understanding circular motion allows pilots to execute precise flight paths and can help predict the forces experienced throughout different aerial maneuvers. Through mastering circular motion, pilots challenge the limits of physical laws safely.