Question

An airplane is dropping an aid package in a remote area. The plane is moving horizontally at speed v0 and the package lands a horizontal distance L from where it was released by the plane. a) Find the time it takes for the package to land. b) Find the altitude of the plane. c) Find the velocity (vector) of the package when it lands. d) Find the speed of the package when it lands. Write your results in terms of v0, L, and g. Check the units/dimensions for each answer.

Answer 1

[a.] Find the time it takes for the package to land.

Vx = ?ax ? Vx = ax(t) + Vo.

x = ?Vx ? x = 0.5ax(t)^2 + Vo(t) + xo.

The initial distance (x0) is 0 and the final distance (x) = L,

? L = 0.5ax(t)^2 + Vo(t)

Manipulating the equation for Vx gives ax = (Vx - Vox)/t.

Plugging this in gives L = 0.5(Vx - Vo)/t (t)^2 + Vo(t)

2L = t(Vx - Vo + 2Vo) ---> t = 2L/(Vx + Vo).

Since we can only give our answer in terms of Vo, L and g,

The final answer is t = 2L/(Vo) because the final velocity will be 0 m/s, at which point the package will have landed.


[b.] Find the altitude of the plane.

Vy = ?ay ---> ay(t) + Voy

Y = ?Vy ---> 0.5ay(t)^2 + Voy(t) + y0

ay = -g and y = 0,

? -0.5g(t)^2 + voy(2L/vox) + y0 = 0

Solving for y0 gives y0 = gL/v0

[c.] Find the velocity (vector) of the package when it lands.

So we're just going to find the velocity in the y direction as well as the x direction

Vy = ay(t) + voy.

voy = 0, and ay = -g,

therefore Vy = [-g(2L/Vox)]j

As for Vx,

Vx = [Vo]i

?V = [Vo]i - [g(2L/Vox)]j


[d.] Find the speed of the package when it lands.

If I remember correctly, speed is just the magnitude of velocity, which would be V = (Vx^2 + Vy^2)^0.5

? V = [(Vo)^2 + (-g(2L/Vox))^2]^0.5

V = [Vo^2 + (4g^2(L^2))/V0x^2))^0.5


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