Does the batter hit the game-winning home run?
Many of the advantages of parametric equations become obvious when applied to solving real-world problems.
Although rectangular equations in x and y give an overall picture of an object’s path, they do not reveal the position of an object at a specific time. This is where your skills in Analytical Trigonometry come in.
A common application of parametric equations is solving problems involving projectile motion.
If an object is thrown with a velocity of v feet per second at an angle of θ with the horizontal, then its flight can be modeled by,
x = (v cos θ ) t and y = v (sin θ ) t – 16 t^2 + h
where t is in seconds and h is the object’s initial height in feet above the ground.
x is the horizontal position and y is the vertical position, and – 16 t^2 represents gravity pulling on the object.
Depending on the units involved in the problem, use g = 32 ft/ s^2 or g 9.8 m/ s^2.
Assume that the ball is hit with an initial velocity of 140 feet per second at an angle of 45°to the horizontal, making contact 3 feet above the ground.
- Find the parametric equations to model the path of the baseball.
- Where is the ball after 2 seconds?
- How long is the ball in the air?
- Is it a home run?
show work and explain your reasoning