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Example 4

Let \begin{array}{rl} f:\left[-\frac{5}{2},\frac{5}{2}\right] & \rightarrow\mathbb{R}^{2}\\ t & \rightarrow\left(t^{3}-4,\, t^{2}-4\right) \end{array}

Then f'(t)=\left(3t^{2}-4,\,2t\right); v(t)=\left|f'(t)\right|=\sqrt{16-20t^{2}+9t^{4}} and the trace of this curve is:

Observe that f(2)=f(-2)=(0,0), which implies that the function is not injective. This means that the particle will be in point (0,0) in two different moments.