Question

Consider a projectile launched at a height h feet above the ground and at an angle θ with the horizontal. If the initial velocity is v₀ feet per second, the path of the projectile is modeled by the parametric equations

x = t(v₀ cos(θ))

and

y = h + (v₀ sin(θ))t − 16t².

The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit h = 3 feet above the ground. It leaves the bat at an angle of θ degrees with the horizontal at a speed of 107 miles per hour (see figure).

(a) Write a set of parametric equations for the path of the ball. (Write your equations in terms of t and θ.)

x	=
y	=

(b) Use a graphing utility to graph the path of the ball when θ = 15°. Is the hit a home run?

YesNo    

(c) Use a graphing utility to graph the path of the ball when θ = 23°. Is the hit a home run?

YesNo    

(d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run. (Round your answer to one decimal place.)
  °

Answer 1

The parametric equations of the position of the ball is given by

Now, the ball is hit from a height h = 3 feet from the ground and with a speed of 107 miles per hour. Converting this speed to feet per second we get,

which we can approximate as 157 feet per second

(a) Hence the parametric equations, with these values are

(b) Now, we set and graph the function

from the graph we can see that the ball touches the ground at 160 feet, so this hit is not a homerun.

(c) Using we have

again, in this case too, the ball hits the ground at roughly 215 feet, so this hit is also not a homerun

(d) Now, using a slider to adjust the value of , we check for when the ball crosses the 400 feet mark at more than 10 feet height, because that is the height of the fence. In this process, we get our answer as 52.57

The corresponding graph is, (the blue line indicates the fence)