Complete Question
A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed at 115 an angle 50 above the horizontal. Is it a home run?
Answer:
[tex]t=5.40sec[/tex]
Step-by-step explanation:
From the question we are told that
Height on hit [tex]H_h=3ft[/tex]
Height of center field fence [tex]H_f=10ft[/tex]
Distance from home plate [tex]D=400ft[/tex]
Initial speed [tex]v_0=115[/tex]
Angle [tex]\theta=50[/tex]
Generally the Parametric equation of the trajectory of a projectile are mathematically represented as
[tex]x=(v_0cos\theta )t[/tex]
[tex]y=(v_0sin\theta )t-\frac{1}{2} gt^2[/tex]
Therefore
[tex]x=(v_0cos\theta )t[/tex]
[tex]x=(115cos50\textdegree )t[/tex]
[tex]x=74t[/tex]
[tex]y=(v_0sin\theta )t-\frac{1}{2} gt^2[/tex]
Initial co_ordinates (0,3)
[tex]y=3+(115sin50\textdegree )t-\frac{1}{2} 32t^2[/tex]
[tex]y=3+88t-16t^2[/tex]
Generally its a home run when Y>10 and x=400
Time when x=400 is
[tex]t=\frac{400}{74}[/tex]
[tex]t=5.40sec[/tex]
