While warming up for a game, a center fielder throws a baseball with a velocity v at different angles above the horizontal.$$ The left fielder always catches the baseball at the same height it was released.$$ Express your answers to the following in terms of the given variables and fundamental constants.

Determine the initial vertical velocity of the baseball.

Determine the total time that the baseball is in the air.

Determine the horizontal displacement of the baseball while it is in the air.$$ Simplify your answer using the mathematical identity $2$sincos $=$ sin$2$$.$

Answer the following questions based on your derivations.

i$.$ What angle$($s$)$ from $0-90$ would give the maximum time of flight?$$ Explain why that makes sense mathematically and conceptually.

ii$.$ What angle$($s$)$ from $0-90$ would give the minimum time of flight?$$ Explain why that makes sense mathematically and conceptually.

iii. Which of the following is true when $=30$ compared to when $=60?$

$\_\_\_$ The ball thrown at $30$ would spend more time in the air

$\_\_\_$ The ball thrown at $60$ would spend more time in the air

$\_\_\_$ The balls thrown at $30$ and $60$ would spend the same amount of time in the air

Explain.

iv$.$ What angle$($s$)$ from $0-90$ would give the maximum horizontal displacement?$$ Explain why that makes sense mathematically and conceptually.

v$.$ What angle$($s$)$ from $0-90$ would give the minimum horizontal displacement?$$ Explain why that makes sense mathematically and conceptually.

vi$.$ Which of the following is true when $=30$ compared to when $=60?$

$\_\_\_$ The ball thrown at $30$ would have more horizontal displacement

$\_\_\_$ The ball thrown at $60$ would have more horizontal displacement

$\_\_\_$ The balls thrown at $30$ and $60$ would have the same horizontal displacement

Explain.