5.3 点の運動

1.

(a) $${{\bf v}(t) = (2 b t - a \pi \sin{\pi t}, \pi a \cos{\pi t} -2 b t )}$$

$${{\bf a}(t) = (-\pi^2 a \cos{\pi t} + 2b, - \pi^2 a \sin{\pi t} - 2b)}$$

$${v = \sqrt{4b^2 + (\pi a - 2b)^2}}$$ $${\hat{{\bf t}} = \frac{(2b,\pi a - 2b)}{\sqrt{4b^2 + (\pi a - 2b)^2}}}$$ $${\hat{\bf n} = \frac{(-\pi^2 a + 2b, -2b)}{\sqrt{4b^2 + (2b - \pi^2 a)^2}}}$$

(b) $${{\bf v}(t) = (3t^2,1 )}$$ $${{\bf a}(t) = (6t,0)}$$ $${v = \sqrt{10}}$$ $${\hat{{\bf t}} = \frac{(3,1)}{\sqrt{10}}}$$ $\hat{\bf n} = (1,0)$

(c) $${{\bf v}(t) = ( 0,2t,2(t-1))}$$ $${{\bf a}(t) = (0,2,2)}$$ $v = 2$ $${\hat{{\bf t}} = (0,1,0)}$$ $${\hat{\bf n} = \frac{(0,1,1)}{\sqrt{2}}}$$

2.

(a) $${\hat{\bf t} = \left(\frac{1}{1 + 2t^2},\frac{2t}{1 + 2t^2},\frac{2t^2}{1 + 2t^2}\right)}$$ $${\kappa = \frac{2}{(1 - 2t^2)}}$$ $${\hat{{\bf n}} = \left(\frac{-2t}{1 + 2t^2},\frac{1 - 2t^2}{1 + 2t^2},\frac{2t}{1 + 2t^2}\right)}$$

$${\hat{\bf b} = \left(\frac{2t^2}{1 + 2t^2},\frac{-2t}{1 + 2t^2},\frac{1}{1 + 2t^2}\right)}$$ $${\tau = \frac{2}{(1 + 2t^2)}}$$

(b) $${\hat{\bf t} = \frac{1}{2}\left(-\sin{t},\cos{t},1\right)}$$ $${\kappa = \frac{1}{2}}$$ $${\hat{{\bf n}} = \left(-\cos{t},-\sin{t},0\right)}$$ $${\hat{\bf b} = \frac{1}{2}\left(\sin{t},-\cos{t},1\right)}$$

$${\tau = \frac{1}{2}}$$