(Figure 1) A relief airplane is delivering a food package to a group of people s

Question

(Figure 1) 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 432 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 . 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? What is the speed vf of the package when it hits the ground?

Explanation / Answer

a) From h = 675 m, the time to impact is T = sqrt(2h/g) = sqrt(2*675/9.8) = 11.74 sec

In that time, the package will travel X = Ux T = (120)*11.74 = 1.4088 miles; so the release point has to be 1408.8 m shy of X marks the spot on the island.

b) We need both the x and y components of the velocity to solve this part. Since we’re ignoring air resistance, the x component will be 432 km/h (120 m/s – see Part B). The y component can be found with the formula v = at. Since the time is 11.74 seconds and acceleration is 9.8 m/s^2, the y component of the velocity is 115.1 m/s. To find the velocity, just use the Pythagorean theorem:

vf = sqrt(vx^2 + vy^2)
vf = sqrt(120^2 + 115.1^2)
vf = 166.28 m/s

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