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 5.5  g
  , or 5.5 times the normal acceleration of gravity, what is the tightest circle (smallest radius) in which the plane can turn?

To solve this problem, we'll use the concept of centripetal acceleration. When an object moves in a circular path, the centripetal acceleration \( a_c \) is given by the formula:

\[
a_c = \frac{v^2}{r}
\]

Where:
- \( v \) is the speed of the object (in this case, the speed of the aircraft),
- \( r \) is the radius of the circular path.

The acceleration the pilot can handle is limited to \( 5.5g \), where \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)). This means the maximum centripetal acceleration \( a_c \) the pilot can withstand is:

\[
a_c = 5.5g = 5.5 \times 9.8 \, \text{m/s}^2 = 53.9 \, \text{m/s}^2
\]

Now, we know the speed of the F-22 Raptor is Mach 1.8, which is 1.8 times the speed of sound (\( 340 \, \text{m/s} \)). So the speed \( v \) is:

\[
v = 1.8 \times 340 \, \text{m/s} = 612 \, \text{m/s}
\]

We can now plug these values into the centripetal acceleration formula and solve for the radius \( r \):

\[
53.9 = \frac{612^2}{r}
\]

Solving for \( r \):

\[
r = \frac{612^2}{53.9}
\]

\[
r \approx \frac{374544}{53.9} \approx 6950.2 \, \text{m}
\]

Thus, the tightest circle (smallest radius) in which the plane can turn without exceeding 5.5g is approximately **6950 meters**.

Would you like further details or have any questions about this solution?

Here are some related questions to consider:
1. What happens if the pilot exceeds 5.5g for a short duration?
2. How does the radius change if the speed of the plane increases to Mach 2.0?
3. How does gravitational acceleration differ for a plane in a horizontal turn versus a dive?
4. What are the physiological effects on a pilot subjected to high g-forces?
5. What technology helps modern fighter jets and pilots cope with high g-forces?

**Tip:** When analyzing circular motion, always remember the relationship between speed, radius, and acceleration.

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 5.5 g, or 5.5 times the normal acceleration of gravity, what is the tightest circle (smallest radius) in which the plane can turn?

Learn how to calculate the smallest turn radius for an F-22 Raptor traveling at Mach 1.8 without exceeding 5.5g. This problem involves key concepts from circular motion, centripetal force, and kinematics. Suitable for high school physics students (Grades 10-12).

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