A projectile is launched from the ground at a $45^\circ$ angle. Its height in feet after $t$ seconds is given by $y(t) = 96t - 16t^2$. Its horizontal displacement in feet is given by $x(t) = 96t$.

1. Write the velocity vector as a function of $t$.

   Hint (a):

   $\vec{v}(t)= \langle x^\prime(t), y^\prime(t) \rangle$

2. Calculate the magnitude of the velocity vector at $t = 0$ and at the moment when the projectile hits the ground. Make a conjecture about the speed with which projectiles returns to earth.

   Step:

   To determine when the projectile hits the ground, solve $0 = 96t - 16t^{2}$.

   Step:

   The velocity vector when $t = 0$ is:

   $\langle 96,96-32(0) \rangle$

   Sketch this vector and use the Pythagorean Theorem to determine its magnitude.