# A pilot flies horizontally at , at height above initially level ground. However, at time t = 0, the pilot begins to fly over ground sloping upward at angle . If the pilot does not change...

A pilot flies horizontally at , at height above initially level ground. However, at time t = 0, the pilot begins to fly over ground sloping upward at angle . If the pilot does not change the airplane's heading, at what time t does the plane strike the ground? I'm not necessarily looking for an answer I'm trying to understand the concept.

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### 1 Answer

We are given say the initial velocity,` v` and height of a plane,` h` flying toward an upward slope at an angle` theta` . (please refer to the attached image)

We have to find how long before the plane hits the slope? At time` t=0` , the plane is at the beginning of the slope, a height, `h` above the level ground. Assuming the plane continues at the same horizontal speed, we wish to find the time at which the plane hits the slope. Given the plane's velocity, the height and the slope's angle, we can relate the horizontal distance to intercept the ramp to the plane's height.

If the plane is at altitude `h` , it will hit the ramp after covering a horizontal distance `d` , where `tan theta=h/d` .

We can relate the horizontal distance to intersect the rampto the plane's altitude using the known slope of ground:

`tan theta=h/d` ...........(i)

Again we know, `d=vt rArr t=d/v` ...........(ii)

Combining (i) and (ii) we get

`t=h/(vtantheta).`

This is the required time it takes for the plane to hit the slope.