Theoretical Mechanics: Fall-2026
HW 2: Due F Day 8 MathBits

1. Zapping with d Zap the following equations with “d”:

   1. \(f= 3x^3-e^{2x}\)

   2. \(g = \sin(y^2)\)

   3. \(5z^4 = \ln(p)\)
2. Separable ODEs

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

   1. \(\frac{dv}{dt}=ct\) where \(v(t=0)=v_0\)

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

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

3. Speed Squared in Curvilinear Coordinates

   Show that the speed squared written in different standard coordinates systems is:

   1. in rectangular coordinates: \(v^2 = \dot{x}^2 + \dot{y}^2+\dot{z}^2\)
   2. in cylindrical coordinates: \(v^2 = \dot{s}^2 + s^2\dot{\phi}^2+\dot{z}^2\)
   3. in spherical coordinates: \(v^2 = \dot{r}^2 + r^2\dot{\theta}^2+r^2\sin^2\theta\dot{\phi}^2\)

   where:

   
   

   The algebra for these calculations is a little tedius, but it's good to do them once in your career to convince yourself that these expressions are true!

   See: GMM: Curvilinear Coordinates for more discussion of curvilinear coordinates. Note that the labels for the aximuthal and polar coordinates in different for math and physics (\(\phi\) and \(\theta\) are swapped.)