A relief airplane is delivering a food package to a group of people stranded on

Question

A relief airplane is delivering a food package to a group of people stranded on a very small island. The island is too small for the plane to land on, and the only way to deliver the package is by dropping it. The airplane flies horizontally with constant speed of 352 km/hour at an altitude of 675 m . The positive x and y directions are defined in the figure. For all parts, assume that the "island" refers to the point at a distance D from the point at which the package is released, as shown in the figure. Ignore the height of this point above sea level. Assume that the acceleration due to gravity is g = 9.80 m/s2 .

A: After a package is dropped from the plane, how long will it take for it to reach sea level from the time it is dropped? Assume that the package, like the plane, has an initial velocity of 352 km/hour in the horizontal direction.

Express your answer numerically in seconds. Neglect air resistance.

B: If the package is to land right on the island, at what horizontal distance D from the plane to the island should the package be released?

Express the distance numerically in meters.

C: What is the speed vf of the package when it hits the ground?

Express your answer numerically in meters per second.

Explanation / Answer

A] Initial Vertical Velocity of the package is 0 m/s.

so, use

h = ut + (1/2)at2

h = 0 + (1/2)(9.8)t2

=> t = (2h/g)1/2 = (2 x 675/9.8)1/2 = 11.737 s

B] The horizontal velocity of the package will not change over time

thus, D = uxt

D = (352 x 5/18)(11.737) = 1147.61 m

C] Vertical velocity of the package at the time it hits the ground will be: vy = u + gt = 0 + 9.8(11.737) = 115.0226 m/s

and horizontal velocity = ux = (352 x 5/18) = 97.777 m/s

therefore the speed vf of the package when it hits the ground will be:

vf = [(115.0226)2 + (97.777)2]1/2 = 150.966 m/s.

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