Question

A bee flies on a trajectory such that its polar coordinate at time t are given by $r=bt/T$²(2T-t) $\theta=t/T$ (0<t<2T) where b and T are positive constants. Find the velocity vector of the bee at time t. Show that the least speed achieved by the bee is b/T. Find the acceleration of the bee at this instant.

Answer 1

$$r=\dfrac{bt}{T^2}(2T-t), \theta=\dfrac{t}{T}, (0<t<2T)$$

$$\dot{r}=\dfrac{2b}{T^2}(T-t), \dot{\theta}=\dfrac{1}{T}$$

$$\ddot{r}=-\dfrac{2b}{T^2}, \ddot{\theta}=0$$

$$v=\dfrac{2b}{T^2}(T-t)\hat{r}+\dfrac{bt}{T^3}(2T-t)\hat{\theta}$$

$$a=(-\dfrac{2b}{T^2}-\dfrac{bt}{T^4}(2T-t))\hat{r}+\dfrac{4b}{T^3}(T-t)\hat{\theta}$$