# MA2051 - Ordinary Differential EquationsSample 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

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.

A
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.

B
Derive from Newton's Law a model of the form .

C
Is the equation in your model linear or nonlinear? Homogeneous or nonhomogeneous?

D
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?

E
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.

A
y'' + 2y' - 8y = 0

B

C
y'' - 8y' + 20y = 0

D

E

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