力学作业（英文版）- (4).pdf

Handout #1
Prof. J. P. How Sept 14, 2004
Due: Sept 28, 2004
Homework Assignment #1
1. Consider a glider flying in a vertical plane at an angle γ to the horizontal. As we
discussed in class, for a glider γ< 0. In a steady glide, the force balance is given by:
D + W sin γ= 0 (1)
L − W cos γ= 0 (2)
The kinematic equations for the system give us that the distance covered along the
ground satisfies x˙= V cos γ and h˙= V sin γ.
For small γ we showed that the flight velocity that gives the flattest glide is given by

2W4 K
Vfg =
ρS CD0
2
where CD = CD0 + KCL.
(a) Show that, for a given height, the distance covered with respect to the ground
satisfies
dx 1
= = −E⇒ R = EΔh
dh γ
where E = CL/CD and R is the range (assume a constant angle of attack so that
E is constant during the glide). What speed does this suggest we should glide at
to maximize the range?
˙˙
(b) Now consider the sink rate hs = −h. Show that

DV 2W C
h˙≈−V γ≈= D
s W ρS 3/2
CL
CD
This suggests that the sink rate is a minimum when 3/2 is a minimum. Show
CL
that this corresponds to a flight speed of

2W4 K
Vms = ≈
ρS 3CD0
CD
(Hint: find the CL that minimizes 3/2 ).
CL
1
Discussion: The conclusion from this analysis is that to maximize range you fly at
the speed that minimizes drag but s