A certain lake is proposed as a landing area for large jet aircraft. The touchdown speed of $100$ mph upon contact with the water is to be reduced to $20$ mph in a distance of $1500$ ft$.$ If the deceleration is proportional to the square of the velocity of the aircraft throug the water, a$=-$k v $^2,$ find the value of the design parameter k$.$ This parameter serves as a measure of the size and shape of the landing gears vanes that plow through the water. Also, determine the time elapsed during the spedified interval. $(1$ space m i l e $=5280$ f t$)$