MA2051 - Ordinary Differential Equations
Sample Exam 2 - B96
Second Exam - Originally Given 1995 D Term
Covers 6.1-6.6, 7.1, 7.2, 8.1, 8.2, 8.4
Instructions: Do your work on the paper provided. Put
your name, section number and PLA's name on that paper. Work
neatly. Show your work. JUSTIFY YOUR ANSWERS.
- 1 40 pts
- Bridge abutments
on many interstate highways are protected by elastic bumpers that
absorb the impact of a collision with a car. This problem asks you to
derive and analyze a simple model of such a system.
See the accompanying figure. For simplicity, suppose the spring
obeys Hooke's Law with constant K and that the car-spring system is
damped by a friction force proportional to velocity. Typically, a car
might have a mass of about M = 1,000 kg and be traveling about 30 m/s
( mph) when it hits the bumper.
- Using the coordinate system indicated in the diagram, write
an expression for the force of the spring on the car when it is
located at position x.
- Derive from Newton's Law a model of the form .
- Is the equation in your model linear or nonlinear?
Homogeneous or nonhomogeneous?
- What range of values of K would guarantee that the system
does not oscillate after a collision? Do these values
correspond to a weaker or a stronger spring?
- Suppose the road is covered with ice so there is no damping
from friction. What value of the spring constant K would limit
the deflection of the car-spring system to 2 m?
- 2 60 pts
- For each of the
following, find a general solution. If initial conditions are given,
solve the initial-value problem. Avoid algebra errors.
- y'' + 2y' - 8y = 0
- y'' - 8y' + 20y = 0
© 1996 by Will Brother.
All rights Reserved. File last modified on December 6, 1996.