How can the quadratic formula be used to determine where a cannonball shot at a known angle and velocity will land?
From physics principles it can be shown that the equation of motion for the vertical displacement
This is a quadratic expression in t and clearly the cannonball will land when its vertical height is zero, that is, when
We may now use the quadratic formula to solve for t in this quadratic equation to obtain
Now in the horizontal (x) direction, the velocity is constant as there is no acceleration in that direction, hence its displacement in the x-direction is given from the definition of velocity as :
Hence it will land a distance