Question

4.4.2 Two Rotations Before Projectile Launch A projectile will be launched from 68° north latitude. Before launch, the longitudinal axis (b) of the projectile is initially aligned with local south (e). by and b are likewise aligned with local east (e) and local vertical (ez), respectively. The projectile is then rotated twice: first, through an azimuth angle of y 45° toward the east next, through an elevation angle of θ 60° up. If the missile's velocity is EvB - 1000 m/sec by find its velocity components in the 1E reference frame.

Answer 1

INTRODUCTION

This is a problem of Euler angles. These are the most commonly used rotational coordinates. In this convention, the reference frame is rotated three times as follows:

1. About z-axis by angle, ψ
2. About y-axis by an angle, θ
3. About x-axis by an angle, φ

For this problem, we require only the first two rotations. The transformation matrices for ψ and θ are as follows:

-sin v cos w 0 0 cos ψ sinh 0

cos θ 0 sin θ -sin θ 0 cos θ

Er Ey E

The transformation matrix A is found by the product of the above two matrices as follows:

cos v cos θ sin ψ _ sin θ -sin v cos v sin θ sin v sin θ cose cos u

STEP 1: Now we need to find the angles ψ and θ. For doing so, we need to define our problem in standard Earth-fixed unit vectors, as follows:

$e_{x}=SOUTH$

e,,-EAST

$e_{z}=UP$

STEP 2: First rotation about z-axis through an azimuth angle of ψ = 45° absolute $e_{z}$

STEP 3: Second rotation about y-axis through an elevation angle of θ = 60° about $-e{}'_{y}$

STEP 4: Velocity calculation by transforming it into E-reference frame by using the above angles in the transformation matrix, as follows:

ET 1000 0