### homework and exercises - Help with Chrstoffel symbols for geometric mechanics problem?

I am working through the book Geometric Control of Mechanical Systems by Bullo and Lewis https://www.amazon.com/gp/product/0387221956/ and I am stuck on a problem, E4-18. The problem was evidently at one point the subject of a research paper, https://pdfs.semanticscholar.org/387d/4bb1c336aa0da87ab1d3a59f53532a2c74d2.pdf . I am trying to reproduce what the authors did in that paper so that I may solve the Bullo and Lewis problem. Taking the paper's authors' kinetic energy function as correct, including their "kinetic energy Riemannian metric", I...Read more

### homework and exercises - Distance travelled by an object slowed only by drag?

Let's say I throw an object horizontally off a cliff with a fixed height, and I know the time it takes to fall. I wanted to know how far it travels, but it has an acceleration opposite the direction of initial velocity due to air resistance. Therefore, I integrated velocity with respect to time; in this case, velocity as a function of time was equal to v0 - at. Okay, what's acceleration (in this case)? I looked into air resistance, and it turns out it's dependent on area, some coefficient, air density, and... instantaneous velocity... As a high...Read more

### homework and exercises - Find the force needed to accelerate body to a certain velocity for a certain time with respect to drag force

So, the problem is straightworward when we suggest that air resistance force is constant:$$\vec F = \frac {\vec V_1 - \vec V_0} {t} m / b$$$$\vec V_0, \vec V_1 - \text {initial and final velocities respectively},\\ t - \text {time, during which velocity will become its final value}, \\b - \text {some constant for drag force}$$ The problem arises when using quadratic air resistance: $$m a = -c v^2$$ I don't have any clue for solving this and asking for help.What I must do is to accelerate plane to the certain velocity in a certain period ...Read more