The runway of an airfield faces west. An airplane, flying in the direction of north-east at a height of 2km and at a speed of 400km/h, on a path which passes over a point 3km west of the runway end, is spotted sqrt(29)km horizontally(south-western quadrant) from the runway end. At this tiem another airplane takes off from the airfield and climbs at an angle of 30degrees with a constant speed of 250km/h.
a) Find the trajectories of both airplanes
b) Determine whether the two planes are in danger of collision.
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To solve this problem, we can break it down into smaller parts and use different concepts such as vector addition, trigonometry, and kinematics.
a) Let's start by finding the trajectory of the first airplane. We know that it is flying in the northeast direction at a height of 2 km and a speed of 400 km/h. From the given information, we can determine the horizontal distance it travels in a specific time.
Using the speed and time, we can find the distance traveled by the first airplane in the northeast direction (in the xy plane). This distance can be calculated using the formula: distance = speed × time.
The time can be found by dividing the horizontal distance of 3 km (west of the runway end) by the speed of the airplane. T = distance / speed = 3 km / 400 km/h = 0.0075 hours.
The horizontal distance traveled by the first airplane can then be calculated as the product of its speed and the time: distance = speed × time = 400 km/h × 0.0075 hours = 3 km.
Therefore, the first airplane has traveled a horizontal distance of 3 km in the northeast direction at a height of 2 km.
Now, let's find the trajectory of the second airplane. We know it takes off from the airfield and climbs at an angle of 30 degrees with a constant speed of 250 km/h. We can use trigonometry to calculate its vertical and horizontal components of motion.
The vertical component (climbing) can be found using the formula: vertical component = speed × sin(angle). Therefore, the vertical component is 250 km/h × sin(30 degrees) ≈ 125 km/h.
The horizontal component (forward motion) can be found using the formula: horizontal component = speed × cos(angle). Therefore, the horizontal component is 250 km/h × cos(30 degrees) ≈ 216.5 km/h.
Therefore, the second airplane climbs at a rate of 125 km/h vertically and moves forward at a speed of 216.5 km/h horizontally.
b) To determine if the two planes are in danger of collision, we need to check if their paths intersect or come close enough to each other.
The position of the first airplane at the time it is spotted is sqrt(29) km horizontally (southwestern quadrant) from the runway end. Since the direction is not specified, let's assume it to be southwest.
The position of the second airplane is at the airfield, which is the starting point of the runway.
To check if the two planes are in danger of collision, we need to determine if their paths intersect. If the horizontal and vertical components of the two airplanes' positions at a particular time are the same, they will collide.
However, from the information provided, we do not have enough details to determine the exact positions of the two airplanes at the same time. Therefore, we cannot conclusively determine if the two planes are in danger of collision.