Mastering Physics Solutions: Delivering a Package by Air

Delivering a Package by Air

Part A = 11.95 s
Part B = 1,389 m
Part C = 369 mph

Solution Below:

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 260 mph at an altitude of 700 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/s^2.

Part A

After a package is ejected from the plane, how long will it take for it to reach sea level from the time it is ejected? Assume that the package, like the plane, has an initial velocity of 260 mph in the horizontal direction.
Express your answer numerically in seconds. Neglect air resistance.

All that matters here is the height from which the package is dropped. The formula is:

x = vit + 1/2at2
700 = 0 * t + 1/2 * 9.8 * t2
700 = 4.9 * t2
t2 = 142.857
t = 11.95 s

11.95 s

Part 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.

First of all, convert mph into meters per second. Mastering Physics will constantly try and catch you with trick questions like this. Be very careful of units. One mph = 0.44704 m/s, so 260 mph = 116.23 m/s.

Now, since the package will be falling for 11.95 seconds and we know the velocity is 260 m/s, just use the formula x = vit + 1/2at2. There is no horizontal acceleration because we ignore air resistance:

x = vit + 1/2at2
x = 116.23 * 11.95 + 1/2 * 0 * 11.95^2
x = 116.23 * 11.95
x = 1,398 m

1,389 m

Part C

What is the speed vf of the package when it hits the ground?
Express your answer numerically in miles per hour.

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 260 mph (116.23 m/s – see Part B). The y component can be found with the formula v = at. Since the time is 11.95 seconds and acceleration is 9.8 m/s^2, the y component of the velocity is 117.11 m/s. To find the velocity, just use the Pythagorean theorem:

vf = sqrt(vx^2 + vy^2)
vf = sqrt(116.23^2 + 117.11^2)
vf = 165 m/s

Now convert to mph. One mph is 0.44704 m/s, so:

Vf = m/s / 0.44704
Vf = 165 / 0.44704
Vf = 369 mph

369 mph

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