Solving for Minimum Angle for Home Run

[jazz20]

Homework Statement

the center field fence in a ball park is 10ft high and 400 feet from home plate. the ball is hit 3 ft above ground. it leaves the bat at an angle of theta degrees with the horizontal speed of 147.67ft/second. find the minimum angle at which the ball must leave the bat in order for hit to be a home run.
the path of the projectile is modeled by the parametric equations:
x=v₀cos(theta)t
y=3+(v₀sin(theta)t-15t^2

Homework Equations

The Attempt at a Solution

substitute 400 in for x and 147.67 for v₀to solve for t
t=400/(146.67cos(theta)
then i took t and put it into the equation for y
y=3+(147.67sin(theta)(400/(146.67cos(theta))-16(400/(146.67cos(theta))^2


Did I do this correctly? If so, can some one help me with the math to solve for theta?

 

[rl.bhat]

1.

The Attempt at a Solution

substitute 400 in for x and 147.67 for v₀to solve for t
t=400/(146.67cos(theta)
then i took t and put it into the equation for y
y=3+(147.67sin(theta)(400/(146.67cos(theta))-16(400/(146.67cos(theta))^2

You can rewrite the equation as
y = 3 + 400*tanθ - 16*400^2*sec^2θ/146.67^2 [ 1/cosθ = secθ]
Put sec^2θ = 1 + tan^2θ,and solve the quadratic for tanθ.

 

[tomcue]

I cannot provide an exact answer to your question without knowing the specific values for t and y. However, I can provide some guidance on how to approach solving for theta.

First, I would recommend using the equation for y to solve for t in terms of theta. This will give you a single equation with only one variable, theta. Then, you can use this equation to find the value of theta that satisfies the condition of the ball clearing the 10ft fence at 400ft away from home plate.

To solve for theta, you can use a numerical or graphical method. For a numerical method, you can use trial and error, plugging in different values for theta until you find the one that satisfies the condition. For a graphical method, you can plot the equation for y as a function of theta and visually determine the minimum angle for the ball to clear the fence.

In general, it is always a good idea to double check your calculations and ensure that all units are consistent. Good luck!