An airplane is flying horizontally with speed 1000 km/h (280 m/s) when an engine falls off. neglecting air resistance, assume that it takes 30 s for the engine to hit the ground. (a) show that the altitude of the airplane is 4.4 km. (use g = 9.8 m/s2.) (b) show that the horizontal distance that the airplane engine travels during its fall is 8.4 km. (c) if the airplane somehow continues to fly as though nothing had happened, where is the engine relative to the airplane at the moment the engine hits the ground

a) y = -1/2 · 9.8 m/s² · (30 s)² = -4410 m = -4.4 km

b) x = 280 m/s · 30 s = 8400 m = 8.4 km

c) The engine is 4.4 km directly below the airplane. Position vector of the engine: (0, -4.4 km)

Explanation:

The equation for the position of the engine is given by the following expression:

r = (x0 + v0x · t, y0 + v0y · t + 1/2 · g · t²)

a)At the final time, the y-component of "r" is the altitude of the plane (ry in the figure).

Then:

y = y0 + v0y · t + 1/2 · g · t²

Since the center of the frame of reference is located where the plane is at the moment the engine falls off, y0 = 0. v0y is also 0. Then:

y = -1/2 · 9.8 m/s² · (30 s)² = -4410 m = -4.4 km

The magnitude of the vector ry is the altitude of the plane, 4.4 km.

b) The horizontal distance traveled by the engine is the magnitude of the rx vector in the figure, the x-component of "r":

x = x0 + v0x · t (x0 = 0 for the same reason as y0)

x = 280 m/s · 30 s = 8400 m = 8.4 km

c) If the airplane continues to fly, the traveled distance of the plane until the engine hits the ground will be:

x = v · t = 280 m/s · 30 s = 8400 m

The position of the engine relative to the airplane is the vector position "re" in the figure. The x-component will be the traveled distance of the plane minus the horizontal distance of the engine flight:

rex = 8400 m -8400 m = 0 m

The vector "rye" is the altitude of the plane, which is the same as the calculated in a).

Relative to the airplane, the engine will be at re = (0, -4.4 km). That is, 4.4 km directly below the airplane.

a displacement distance is size of displ

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