Math 1320 - Lab 9
uNiD:
Name:
Due: 11/07
Today we will continue with some application problems in vectors. Don’t forget to write your reasoning (in
COMPLETE and CONCISE sentences) behind your approach to each problem so that someone who has
taken Math 1310 would understand what you are doing.
1. The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates
ρ, θ, φ, as follows. We take the origin to be the center of the earth and the positive z-axis to pass
through the North Pole. The positive x-axis passes through the point where the prime meridian (the
meridian through Greenwich, England) intersects the equator. Then the latitude of P is α = 90o − φo
and the longitude is β = 360o − θo . Find the great-circle distance from Los Angeles (lat. 34.06o N, long.
118.25o W) to Montreal (lat. 45.50o N, long. 73.60o W). Take the radius of the earth to be 3960 mi. (A
great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)
2. In solving the equations governing the motion of 1-dimensional shallow water waves (the Kortewegde
Vries equations a.k.a KdV equations), we have a solution that gives wave height, h, as a function of
space, x, and time, t.
h(x, t) = 12
A space-time plot is shown below
3 + 4 cosh(2x − 8t) + cosh(4x − 64t)
(3 cosh(x − 28t) + cosh(3x − 36t))2
Sketch what the wave looks like at t = −0.2 and t = 0.2. Imagine that there is a wave height probe at
x = −2. What would a data recording from this probe look like for −0.5 ≤ t ≤ 0.5? Describe what is
happening in this system in your own words.
3. The United States Army have been involved in missile defense systems since the start of the cold war.
Recently, new systems called Terminal High Altitude Area Defense (THAAD) have been developed.
Their main aim is to shoot down short, medium and intermediate ballistic missiles.
The contractors on this project need to test a physical prototype of this system so they decide to conduct
a test in Cape Canaveral, Florida. They use a dummy ballistic missile and an interceptor missile with
paths defined by the space curves, ~r1 (t) and ~r2 (t) respectively.
~r1 (t) = ht, t2 , t3 i
~r2 (t) = h1 + 2t, 1 + 6t, 1 + 14ti
(a) Does the interceptor missile successfully take down the dummy ballistic missile (i.e do the missiles
collide)?
(b) Do the paths of the two missiles intersect?
(c) Comment on the missile test and suggest what needs to be tweaked in the system based on what
you found in (a) and (b).