A particle of mass m moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:
F=+m \alpha^{2} x
In simple harmonic motion, the position of the particle as a function of time can be written
x(t)=A \cos \omega t+B \sin \omega t
Likewise, in the present case we have
x(t)=A f_{1}(t)+B f_{2}(t)
for some appropriate functions f_{1} and f_{2},
a. f_{1}(t) and f_{2}(t) can be chosen to have the form e^{r t}. What are the two appropriate values of r ?
b. Suppose that the particle begins at position x(0)=x_{0} and with velocity v(0)=0. What is x(t) ?
c. A second, identical particle begins at position x(0)=0 with velocity v(0)=v_{0}. The second particle becomes closer and closer to the first particle as time goes on. What is v_{0} ?