Please help with this math problem

Question

If a punter kicks a football with an upward velocity of 80 feet per second and his foot meets the ball 2 feet off the ground, the function h(t)= -16t^2+80t+2 represents the height of the ball, h(t), in feet, after t seconds

Answer 1

h(t)=-16t²+80t+2 is the equation of motion. When t=0 h is 2 feet which confirms that the ball meets the punter’s foot two feet from the ground initially.

h(t)=2-16(t²-5t+25/4-25/4) is another way of writing the same equation.

So h(t)=2-16((t-5/2)²-25/4)=2+100-16(t-5/2)²=102-16(t-5/2)² is another way of writing the equation that tells us something about how the ball moves. It tells us that when t=5/2=2.5 seconds the ball reaches its maximum height of 102 feet, because the term 16(t-5/2)² becomes zero. At any other value of t the height is reduced from 102 feet. It also tells us that when 16(t-5/2)²=102, the ball reaches the ground. We can solve this equation: t-5/2=√(102/16), t=5/2+√(51/8)=2.5+2.5249=5.0249 seconds approx when the ball falls to the ground. Note that the ball is 2 feet off the ground again when 16(t-5/2)²=100, that is when t=5/2+√(100/16)=5/2+5/2=5 seconds.