EML 5060 Analysis in Mechanical Engineering 9/22/99
Closed book Van Dommelen 10:45-11:35am

Show all reasoning and intermediate results leading to your answer, or credit will be lost. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator.

1.

A car is cornering according to the curve

y² = 1 + x²

(x and y>0 in meters) with a constant velocity of two m/sec. Draw the path that the car travels, showing all features clearly. Find the velocity components for any car position x. Solution.

2.

Various problems, such as finding the period of a pendulum that takes large swings, involve integrals that are nontrivial, but that can be evaluated using approximation. As a simple example, evaluate

to an error that is guaranteed to be less than 0.00003. Do this using only the +, -, *, and / operations on your calculator. Solution.

3.

We want to shoot a directed laser beam weapon through an opening at a distant enemy. In order to have the best chance of hitting within the enemy's rank, we must shoot in a direction parallel to the line x+y+z=2, x+2y+3z=4. Find a vector in the direction of this line. If the opening through which we must shoot is at the location x=2, y=1, and z=3, and we are standing on level ground, (being the plane z=0), give an equation for the desired line of fire and the location (x,y) from which we must fire our weapon. Solution.