Difficult# Relative Position Vectors: A Crow and a Hawk

CLMECH-KEANEG

A hawk is chasing a young crow. The positions of the hawk and crow as functions of time are described by the following vectors, respectively:

$$\vec {r}_h = (t^2 - 14 \space t +45) \cdot \hat {i} + (t \space – \space 9) \cdot\hat {j} + t \cdot\hat {k}$$

and

$$\vec {r}_c = (2t^2 -19 \space t +9) \cdot \hat {i} +(t^2 - 11 \space t +18) \cdot \hat {j} + t \cdot\hat {k}$$

...where $\hat {i}, \hat {j}$ and $\hat {k}$ are the unit vectors in the $xyz$ coordinate system and all the units are SI. The subscript $h$ refers to the hawk and the subscript $c$ to the crow.

The crow can reach the safety of its nest (and its mother's protection) in 8 s.

Will the young crow escape the hawk? If not, after how many seconds will the hawk catch the crow? If yes, how far away from the nest will the hawk be when the young crow enters its nest?