One of the limiting factors in high-performance aircraft is the number of g 's to which the pilot can be subjected without blacking out. The new F−22 Raptor fighter can achieve Mach 1.8 (1.8 times the speed of sound, which is about 340 m/s). Suppose a pilot dives in a circle and pulls up. If the pilot cannot exceed 6.0  g  , or 6.0 times the normal acceleration of gravity, what is the tightest circle (smallest radius) in which the plane can turn?
Express your answer using two significant figures.

To determine the smallest radius of the circular path the pilot can take without exceeding 6.0 g, we need to relate the centripetal acceleration of the plane to the gravitational acceleration limit.

### Key Equations:
The centripetal acceleration for an object moving in a circular path is given by:
\[
a_c = \frac{v^2}{r}
\]
Where:
- \( a_c \) is the centripetal acceleration,
- \( v \) is the velocity of the aircraft,
- \( r \) is the radius of the circular path.

The pilot's maximum allowable acceleration is 6.0 g, where \( g \) is the acceleration due to gravity (\( g = 9.8 \, \text{m/s}^2 \)). Thus, the maximum allowable centripetal acceleration is:
\[
a_c = 6.0 \times 9.8 \, \text{m/s}^2 = 58.8 \, \text{m/s}^2
\]

Next, we need the velocity \( v \) of the aircraft, which is given as Mach 1.8. Since the speed of sound is approximately 340 m/s, the velocity is:
\[
v = 1.8 \times 340 \, \text{m/s} = 612 \, \text{m/s}
\]

Now, using the centripetal acceleration formula, we can solve for the radius \( r \):
\[
a_c = \frac{v^2}{r} \implies r = \frac{v^2}{a_c}
\]

Substitute the known values for \( v \) and \( a_c \):
\[
r = \frac{(612 \, \text{m/s})^2}{58.8 \, \text{m/s}^2} = \frac{374544}{58.8} \approx 6368.4 \, \text{m}
\]

### Final Answer:
The tightest circle (smallest radius) in which the plane can turn is approximately:
\[
r \approx 6400 \, \text{m}
\]
to two significant figures.

Would you like more details or have any questions?

### Related Questions:
1. How does increasing the speed of the plane affect the radius of the turn?
2. What happens to the pilot's experience when the number of g’s exceeds 6.0?
3. How do different altitudes affect the speed of sound, and consequently the aircraft's velocity?
4. What is the significance of centripetal force in aircraft maneuvers?
5. How does pilot training prepare individuals for high g-forces?

**Tip:** Remember that reducing speed can help an aircraft make tighter turns without exceeding g-force limits.

One of the limiting factors in high-performance aircraft is the number of g's to which the pilot can be subjected without blacking out. The new F−22 Raptor fighter can achieve Mach 1.8 (1.8 times the speed of sound, which is about 340 m/s). Suppose a pilot dives in a circle and pulls up. If the pilot cannot exceed 6.0 g, or 6.0 times the normal acceleration of gravity, what is the tightest circle (smallest radius) in which the plane can turn? Express your answer using two significant figures.

Explore how to calculate the tightest turn radius an F-22 Raptor can make at Mach 1.8 without exceeding a 6.0 g-force limit. This problem demonstrates the application of centripetal force and acceleration principles in circular motion, suitable for physics students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation