A pilot performs an evasive maneuver by diving vertically at a constant 310 m/s. If he can withstand an acceleration of 9.0 g’s without blacking out, at what altitude must he begin to pull his plane out of the dive (moving in a vertical circular path) to avoid crashing into the sea?

Verified step by step guidance

1. Identify the key variables in the problem: the speed of the plane \( v = 310 \; \text{m/s} \), the maximum acceleration the pilot can withstand \( a = 9.0g = 9.0 \times 9.8 \; \text{m/s}^2 \), and the goal is to find the minimum radius of the circular path \( r \) to avoid crashing into the sea.
2. Recall the formula for centripetal acceleration in circular motion: \( a_c = \frac{v^2}{r} \). Here, \( a_c \) is the centripetal acceleration, \( v \) is the speed, and \( r \) is the radius of the circular path.
3. Rearrange the formula to solve for the radius \( r \): \( r = \frac{v^2}{a_c} \). Substitute \( a_c = 9.0 \times 9.8 \; \text{m/s}^2 \) and \( v = 310 \; \text{m/s} \) into the equation.
4. Calculate the radius \( r \) using the substituted values. This radius represents the minimum distance from the center of the circular path to the plane. Since the plane is diving vertically, the altitude at which the pilot must begin pulling out of the dive is equal to this radius.
5. Interpret the result: The altitude at which the pilot must begin to pull out of the dive is equal to the calculated radius \( r \). This ensures the centripetal acceleration does not exceed the maximum tolerable acceleration of \( 9.0g \), preventing the pilot from blacking out and avoiding a crash into the sea.

Key Concepts

Acceleration and G-Forces

Acceleration is the rate of change of velocity of an object. In this context, the pilot can withstand an acceleration of 9.0 g's, where 'g' represents the acceleration due to Earth's gravity (approximately 9.81 m/s²). This means the pilot can endure an acceleration of about 88.29 m/s² before experiencing blackout, which is crucial for determining the safe altitude for the maneuver.

Vertical Circular Motion

When an object moves in a vertical circular path, it experiences varying forces due to gravity and centripetal acceleration. As the pilot pulls out of the dive, the plane must generate enough lift to counteract the gravitational force and provide the necessary centripetal force to maintain the circular motion. Understanding the dynamics of vertical circular motion is essential for calculating the required altitude.

Kinematics and Energy Conservation

Kinematics involves the study of motion without considering the forces that cause it. In this scenario, energy conservation principles can be applied, where the kinetic energy of the plane at the bottom of the dive must be converted into potential energy as it ascends. This relationship helps determine the altitude needed to safely pull out of the dive without crashing.