Download HW 2: Reference frames and coordinate systems

Aerospace Engineering 2220: Dynamics
Prof. Eric Feron
HW 2
Due September 9, 2016
1
Collision between two aircraft
Consider two aircraft present on the same runway, and moving in opposite
directions, as shown in Fig. 1. The two aircraft are initially 5 km away from
each other. The first aircraft, labeled 1, is taxiing at low speed (about 40 km
per hour). The other aircraft, labeled 2, is taking off: starting from 0 initial
speed, it is accelerating at constant acceleration equal to 0.5 m/sec2 .
Figure 1: Two aircraft on a collision course
1. Write an expression for the distance between the two aircraft as a
function of time.
2. Let O be the point where the two airplanes collide. What is the location
of O? (Assume each aircraft is a point).
3. Compute analytical formulas for the speeds and accelerations of both
aircraft in the reference frame (O, i, j) and the basis (i, j) as shown.
What are the values of these speeds and accelerations at the time of
collision?
4. Repeat Question 3, but use the reference frame (O1 , i1 , j1 ) and the basis
(i1 , j1 ). Repeat the same question with the reference frame (O2 , i2 , j2 )
and the basis (i2 , j2 ).
5. Compute analytical formulas for the speed and acceleration of aircraft
2 in the reference frame (O1 , i1 , j1 ) of aircraft 1, but using the basis
(i, j). What are the speed and accelerations at the time of collision?
2
Earth, sun, and moon
In this problem, we compute formula for the motion of the sun, the earth,
and the moon in each other’s reference frames. In order to make your life
bearable, we will make gross simplifications. In this problem, we assume we
have a ”fixed” reference frame. That reference frame is composed of: The
center of the sun, and three independent vectors, each one pointing to a
really far-away star. In that ”fixed” reference frame, we have the following
information:
1. The sun rotates around itself in about 25 days (this is an approximation).
2. The earth rotates around itself in 23 hours and 56 minutes.
3. The Earth is about 152 million km away from the sun.
4. The Earth makes a full orbit around the sun in 365.256 days.
5. The moon orbits the Earth and around itself in 27.322 days.
6. The moon is 363,104 km away from the Earth’s center.
We assume that Earth, Moon, and Sun motions are all in the same plane,
and that the axes of orbits and rotation are all orthogonal to that plane. A
Youtube animation of the motion is in https://www.youtube.com/watch?v=W47Wa7onrIQ
for you to read. We also assume that at time t = 0, the Sun, Earth, and
Moon are all aligned, in that order.
1. What are the speed and acceleration of the Sun in the reference frame
of the Earth? (Write down analytical expressions)
2. What are the speed and acceleration of the moon in the ”fixed” reference frame? In the Sun’s reference frame?
3. What are the speed and acceleration of the moon in the Earth’s Reference frame? In the Sun’s reference frame?
3