2 Dimensional Motion-Kinematics
QS 1: An airplane wishes to travel 800 km northeast of a starting point. There is a wind blow towards the east at 100km/h. Assuming the maximum speed of the plane in still air is 500km/h,
a: what course should the pilot steer to arrive in minimum time?
b: How long does the trip take?
n = speed north
n - 100 = speed east
n^2 + (n - 100)^2 = 500^2
2 n^2 - 200n + 10000 - 250000 = 0
n^2 - 100 n - 120000 = 0
solve for n
the north distance is 400√2
time = distance / speed
the bearing (Θ , east of north)
... tan(Θ) = (n - 100) / n
a. Vp + 100 = 500[45o].
Vp + 100 = 500*sin45 + i500*Cos45,
Vp + 100 = 353.6 + 353.6i,
Vp = 253.6 + 353.6i = 435km/h[35.6o]E. of N.
Tan A = X/Y.
b. d = V*T.
800 = 435 * T,
=
To solve this problem, we need to consider the vector components of the plane's velocity and the effects of the wind. Let's break down the steps to find the answers to both parts of the question:
a) To find the course the pilot should steer to arrive in minimum time, we need to consider the combined effect of the plane's velocity and the wind velocity.
1. Determine the plane's velocity vector:
The plane's maximum speed in still air is 500 km/h. Since we want to travel in the northeast direction, which is halfway between east and north, we can divide the plane's speed equally in both directions. Therefore, the plane's velocity relative to the ground would be 500 km/h in the northeast direction.
2. Calculate the wind's velocity vector:
The wind is blowing towards the east at 100 km/h. Since we're traveling in the northeast direction, we need to take the wind component in that direction as well. Using trigonometry, we can determine that the eastward component of the wind is 100 km/h multiplied by the cosine of 45 degrees (as northeast is a 45-degree angle from both east and north). So, the wind's eastward velocity component is 100 km/h * cos(45).
3. Determine the resultant velocity:
To find the resultant velocity, we need to add the plane's velocity vector to the wind's velocity vector. The resultant velocity vector represents the actual velocity of the plane relative to the ground.
To add the vector components, we'll consider the eastward direction as the positive x-axis and the northward direction as the positive y-axis.
The eastward component of the resultant velocity will be the sum of the plane's eastward velocity and the wind's eastward velocity. The northward component will be the sum of the plane's northward velocity and the wind's northward velocity (which is zero in this case since the wind does not blow north).
4. Determine the course angle:
The course angle is the angle between the resultant velocity vector and the east direction. We can calculate this angle using trigonometry. Using the inverse tangent function, we can find the angle as atan(northward component / eastward component).
b) To find the time it takes for the trip, we need to find the magnitude of the resultant velocity vector and divide the distance by that magnitude.
1. Calculate the magnitude of the resultant velocity:
The magnitude of the resultant velocity is the speed at which the plane travels relative to the ground. We can use the Pythagorean theorem to find the magnitude, which is the square root of the sum of the squares of the components.
2. Calculate the time taken:
Divide the distance traveled (800 km) by the magnitude of the resultant velocity to determine the time taken for the trip.
By following these steps, you should be able to find the answers to both parts of the question.