This question is asked on my differential topology mock mid-term, but I can't figure out what to do:
Consider smooth curves $\gamma_i: \mathbb{R} \to \mathbb{R}^2, i = 1, . . . , n$ which represent the trajectories of $n$ moving obstacles (this means at time $t$ the obstacle $i$ is at $\gamma_i(t)$). Assume that $\gamma_i(0) \neq (0, 0)$ for all $i = 1, . . . , n$.
Let $\epsilon > 0$, and $P \in \mathbb{R}^2$. Prove that there exists a trajectory on which a projectile can be send so that:
• it starts at time $t = 0$ from the point $(0, 0)$,
• it moves with constant speed, along a straight line,
• at time $t = 1$, its distance to $P$ is smaller than $\epsilon$
• and the projectile avoids all obstacles.
My start was as following: We are looking for $\gamma: \mathbb{R} \to \mathbb{R}^2$, $\gamma(0) = (0, 0)$. Now we need to find some $x = (x_1, x_2) \in B_\epsilon(P)$ to satisfy the last condition, because obviousy we will then set $\gamma(t) = tx$ with $d\gamma_t = (x_1, x_2)$ for all $t$.
We will then also have $\gamma(1) = x \in B_\epsilon(P)$.
I have no clue however how to find $x$. Neither do I see the connection to most of the subjects we covered in the lectures (Chapter $1$ from Guillemin and Pollack).
We, the class, have figured this out ourselves. I' ll post a rough outline here, the details of which are easily filled in.
For every $t \neq 0$, the point $\gamma_i(t) = (a, b)$ for some $(a, b) \in \mathbb{R}^2$ can be transformed to $(\frac{a}{t}, \frac{b}{t})$. This gives us $n$ new curves, $\frac{\gamma_i}{t}$.
If we look at $B_{\epsilon}(P) \cap \cup_i \frac{\gamma_i}{t}$ we see that this has measure zero, because all our curves have measure zero and a finite union of these will still have measure zero. So by Sard's theorem we can pick a point $(a, b) \in B_{\epsilon}(P)$, $ (a, b) \notin B_{\epsilon}(P) \cap \cup_i \frac{\gamma_i}{t}$. If we now set $\gamma(t) = t(a, b)$ then this will satisfy all our conditions:
Because if $\gamma(t) = \gamma_i(t)$ for some $i$, then it would follow that $t \neq 0$ and thus $\frac{\gamma(t)}{t} = \frac{\gamma_i(t)}{t}$ which implies $(a. b) \in B_{\epsilon}(P) \cap \cup_i \frac{\gamma_i}{t}$, thus contradiction.