Final Exam Review: Airplane

Metamorphose

1. Using a reference frame with the origin at the take-off airport, the positive x-axis due East,
and the positive y-axis due North, the acceleration a of an airplane as a function of time
can be described as:
a = (αt)(i) + (βt⁴ - γt)(j)

with α, β, and γ positive and constant.
Assuming that the airplane takes off from the airport at time t = 0 with zero initial
velocity:
a) What are the units of α, β, and γ?
b) Find the time(s) when the airplane position is directly NE of the airport.
c) Find the trajectory of the plane, y(x).
Write your results in terms of α, β, and γ. Remember to check the dimensions/units for each
answer.

2. Relevant equations

[b] a = (αt)(i) + (βt⁴ - γt)(j)

v = ∫a(dt) → (0.5αt²)(i) + (1/5(βt⁵) - 0.5γt²)(j)

r = ∫v(dt) → (1/6(αt³)(i) + (1/30(βt⁶ - 1/6(γt³)(j)

3. The attempt at a solution

[a.] For part A,

the units for α, β, and γ respectively:

ms^-3, ms^-6, and ms^-3

[b.] For part B, to find the times at which the plane is directly NE would mean setting the r_x and r_y components equal to each other.

∴r_x = r_y

1/6(αt³ = 1/30(βt⁶ - 1/6(γt³

1/6(αt³ + 1/6(γt³ = 1/30(βt⁶

Multiplying the entire thing by 30 gives:

5αt[SUP]3[/SUP + 5γt³ = βt⁶

t³(5α + 5γ) = βt⁶

(5α + 5γ) = βt³

t³ = (5α + 5γ)/β

∴t = [(5α + 5γ)/β]^1/3

[c.] For part C, I know trajectory means projectile motion, but I am not sure how to do it.

Do we simply get the time and plug it into r_y?


Doing that gives 5α(α - γ)/6β as a final answer.