Theoretical Mechanics: Fall-2021
Practice Separable ODEs: Due Day 12

1. Separable ODEs Solve the following differential equations by separation. \(b\) and \(c\) are constants with the appropriate dimensions.

   1. \(\frac{dv}{dt}=ct\) where \(v(t=0)=55\)

   2. \(\frac{dp}{dm}=-cp\) where \(p(m=0)=p_0\)

   3. \(\frac{dz}{dk}=-b-cz\) where \(z(k=0)=z_0\)

   4. \(\frac{df}{dg}=-b-cf^2\) where \(f(g=0)= 0\)

      This one is way harder and you might not be able to do it for a couple of weeks.

2. Constant Acceleration by Separation of Variables

   1. Calculate: Treat Newton's 2nd law as a separable differential equation and solve for the velocity and position as a function of time of an object

      • moving in 1-D
      • is not initially at the origin of coordinates,
      • is moving with a non-zero speed,
      • and experiences a constant force.

   2. Reflect: Do your answers look familiar? If yes, from where? If not, how would you have to modify these equations to be similar to equations you know?