an airplance traveling at 450 km/h needs to reverse its course. the pilot decides to bank the wings at 12 degrees. find the time needed to reverse the course
force in = mg sin 12 = m v^2/r
v^2/r = g sin 12
turns 180 deg = pi radians
r = v^2/(g sin 12)
pi r = distance flown = pi v^2/(g sin 12)
time = distance / speed = pi r/v
= pi v/(g sin 12
here v = 450/3.6 and g = 9.81
To find the time needed to reverse the course, we need to analyze the forces acting on the airplane during the banked turn. The first step is to determine the radius of the turn.
In a banked turn, two forces are involved: the gravitational force acting downwards and the centripetal force acting towards the center of the turn.
We can start by calculating the centripetal force using the formula:
Centripetal force = (mass of the airplane) x (centripetal acceleration)
The centripetal acceleration can be derived from the formula:
Centripetal acceleration = (velocity^2) / (radius)
Here, the centripetal acceleration is equal to the gravitational force acting on the plane:
Gravitational force = (mass of the airplane) x (gravitational acceleration)
The gravitational acceleration on Earth is approximately equal to 9.8 m/s^2.
Next, we need to convert the given velocity from km/h to m/s:
450 km/h = 450,000 m/3600 s ≈ 125 m/s
Since we are dealing with high-speed aircraft, we can assume that the airplane's weight is balanced by the lift force perpendicular to the wings.
Now, let's calculate the angle between the vertical axis and the plane's normal reaction force, as follows:
tan(θ) = (centripetal acceleration) / (gravitational acceleration)
tan(θ) = (125^2) / (radius x 9.8)
Now, we can find the radius of the turn:
radius = (125^2) / (tan(12) x 9.8)
Lastly, to find the time needed to reverse the course, we divide the distance traveled in half by the velocity:
time = (distance traveled) / (2 x velocity)
distance traveled = 2 x π x radius
Plug in the calculated radius and the velocity of 125 m/s to find the time needed to reverse the course.