Peregrine falcons are known for their maneuvering ability. In a tight circular turn, a falcon can attain a centripetal acceleration 1.5 times the free-fall acceleration. What is the radius of the turn if the falcon is flying at 25 m/s?

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Identify the given values: the centripetal acceleration is 1.5 times the free-fall acceleration, so \( a_c = 1.5g \), where \( g = 9.8 \ \text{m/s}^2 \). The velocity of the falcon is \( v = 25 \ \text{m/s} \). The goal is to find the radius of the turn \( r \).

Recall the formula for centripetal acceleration: \( a_c = \frac{v^2}{r} \), where \( a_c \) is the centripetal acceleration, \( v \) is the velocity, and \( r \) is the radius of the turn.

Substitute the expression for \( a_c \) into the formula: \( 1.5g = \frac{v^2}{r} \).

Rearrange the equation to solve for \( r \): \( r = \frac{v^2}{1.5g} \).

Substitute the known values of \( v = 25 \ \text{m/s} \) and \( g = 9.8 \ \text{m/s}^2 \) into the equation to calculate \( r \): \( r = \frac{(25)^2}{1.5 \times 9.8} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that keeps an object moving in that path. It is calculated using the formula a_c = v^2 / r, where v is the tangential speed and r is the radius of the circular path. In this scenario, the falcon experiences a centripetal acceleration that is 1.5 times the acceleration due to gravity, which is approximately 9.81 m/s².

Free-Fall Acceleration

Free-fall acceleration, often denoted as g, is the acceleration experienced by an object due solely to the force of gravity acting on it, typically measured as 9.81 m/s² on the surface of the Earth. This value is crucial for comparing other accelerations, such as the centripetal acceleration experienced by the falcon during its maneuver. Understanding this concept helps in determining the forces acting on the falcon during its flight.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle or a circular path. In this context, the falcon's flight involves uniform circular motion, where it maintains a constant speed while changing direction. The relationship between speed, radius, and centripetal acceleration is fundamental in analyzing the dynamics of circular motion, allowing us to calculate the radius of the turn based on the given speed and acceleration.

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