a quarterback throws a football in a high, arching spiral to a receiver running down the field. where is the football traveling the fastest? where is it traveling the slowest?

Travelling fastest: when the football leaves the hands of the quarterback and when the receiver gets it

Travelling slowest: at the very top of then arched trajectory

Explanation:

Notice that this example can be analyzed with a two dimensional pattern as in the launching of a cannonball, where we study separately the velocities and acceleration acting vertically and horizontally.

The football is thrown at an agle with the horizontal with an initial velocity that is decomposed in the vertical and in the horizontal axes.

The Horizontal movement of the football is that of an object with constant velocity (that of the horizontal component of the initial velocity imparted by the quarterback) .

The vertical movement of the football is that of an object moving in an accelerated fashion, with constant acceleration due to the gravitational field, and with an initial velocity opposite to that constant acceleration. The initial velocity is that of the vertical component of the initial velocity imparted by the quarterback.

AT every point on the path of the ball, its velocity can be calculated by the vector addition of the horizontal component (with constant velocity as we discussed above) and the vertical component (this velocity is changing since its is under accelerated motion) of the object's velocity at any given time through the path.

The maximum velocity of the football will be at the point where the two components are at their maximum (that is when the football leaves the hand of the thrower, and when it gets to the hands of the receiver.

While the minimum is going to be when the vertical component of the velocity is at is minimum (zero) at the top of the arched trajectory when the football changes vertical direction and starts heading downwards.