Answer:
To determine the gauge pressure at the stagnation point on the nose of the plane, we need to consider the Bernoulli's principle, which states that the pressure of a fluid decreases as its velocity increases.
First, we need to convert the speed and altitude to SI units:
Speed of the plane: 300 km/h = 83.33 m/s
Altitude: 12,000 m
To solve the problem, we need to consider two pressure components: the atmospheric pressure at the given altitude and the dynamic pressure due to the speed of the plane.
1. Speed of 300 km/h:
At an altitude of 12,000 m, the atmospheric pressure can be determined using the barometric formula or atmospheric pressure tables. Let's assume the atmospheric pressure at this altitude is 22,000 Pa.
Since the speed of the plane is relatively low, we can neglect the dynamic pressure component for this scenario. Therefore, the gauge pressure at the stagnation point would be equal to the atmospheric pressure, which is 22,000 Pa.
2. Speed of 1050 km/h:
Following the same process, let's assume the atmospheric pressure at an altitude of 12,000 m is still 22,000 Pa.
Considering the higher speed of the plane, we cannot neglect the dynamic pressure component in this scenario.
The dynamic pressure, q, can be calculated using the equation: q = (1/2) * ρ * V^2, where ρ is the air density and V is the velocity of the plane.
At an altitude of 12,000 m, the air density ρ is approximately 0.382 kg/m^3 (assuming standard atmospheric conditions).
Converting the speed to m/s: 1,050 km/h = 291.67 m/s.
Substituting the values into the dynamic pressure equation, we find q ≈ 20,530 Pa.
To determine the gauge pressure, we need to add this dynamic pressure to the atmospheric pressure: 22,000 Pa + 20,530 Pa = 42,530 Pa.
Therefore, if the speed of the plane were 1050 km/h, the gauge pressure at the stagnation point on the nose of the plane would be approximately 42,530 Pa.