Does The Batter Hit The Game-Winning Home Run? ✓ Solved

Does The Batter Hit The Game Winning Home Runmany Of The Advantages O

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. Determine the trajectory and final position of the baseball based on the given initial conditions and analyze whether it results in a home run.

Sample Paper For Above instruction

The trajectory of a baseball hit with an initial velocity of 140 feet per second at an angle of 45° to the horizontal, starting 3 feet above the ground, can be modeled using parametric equations. These equations are derived from the basic physics of projectile motion, considering the effects of gravity and initial velocity components in the horizontal and vertical directions.

Step 1: Decompose initial velocity into components:

Initial velocity, v = 140 ft/sec

Angle, θ = 45°

Horizontal component: v_x = v cos θ = 140 cos 45° = 140 (√2 / 2) ≈ 140 0.7071 ≈ 99.0 ft/sec

Vertical component: v_y = v sin θ = 140 sin 45° = 140 * (√2 / 2) ≈ 99.0 ft/sec

Step 2: Write parametric equations:

Horizontal position (x): x(t) = v_x * t = 99.0 t

Vertical position (y): y(t) = h + v_y * t - 16 t² = 3 + 99.0 t - 16 t²

Here, t is in seconds, x and y are in feet.

Step 3: Find the position after 2 seconds:

x(2) = 99.0 * 2 = 198 ft

y(2) = 3 + 99.0 2 - 16 (2)² = 3 + 198 - 16 * 4 = 3 + 198 - 64 = 137 ft

The baseball after 2 seconds is approximately at (198 ft, 137 ft).

Step 4: Determine the total time in the air:

To find when the ball hits the ground (y=0), solve y(t) = 0:

0 = 3 + 99.0 t - 16 t²

Rewrite as: 16 t² - 99.0 t - 3 = 0

Apply quadratic formula:

t = [99.0 ± sqrt(99.0² - 4 16 (-3))] / (2 * 16)

t = [99.0 ± sqrt(9801 + 192)] / 32

t = [99.0 ± sqrt(9993)] / 32

t ≈ [99.0 ± 99.96] / 32

This gives two solutions:

  • t ≈ (99.0 + 99.96)/32 ≈ 198.96 /32 ≈ 6.22 seconds
  • t ≈ (99.0 - 99.96)/32 ≈ -0.96 /32 ≈ -0.03 seconds (discard negative time)

Therefore, the ball hits the ground approximately after 6.22 seconds.

Step 5: Determine if it's a home run:

A typical baseball field has the outfield fence approximately 400 feet away from home plate. The horizontal distance traveled at t = 6.22 seconds is x(6.22) ≈ 99.0 6.22 ≈ 615.78 feet. Since 615.78 feet exceeds the 400 feet distance, and the ball's maximum height can be found at t = v_y / (2 gravity) = 99.0 / (2 * 16) ≈ 3.09 seconds, where y is maximized. Plugging t in the y(t):

y(3.09) = 3 + 99.0 3.09 - 16 (3.09)² ≈ 3 + 305.91 - 16 * 9.55 ≈ 3 + 305.91 - 152.8 ≈ 156.11 feet

The maximum height exceeds typical outfield fence height (around 10 to 15 feet). Given the horizontal extent of approximately 616 feet, well beyond the 400 feet home run distance, this indicates a clear home run.

In conclusion, based on the initial velocity and angle, the baseball reaches a flight long enough and high enough to go beyond a typical outfield fence, confirming it is a home run.

References

  • Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers with Modern Physics (9th ed.). Brooks Cole.
  • Hewitt, P. G. (2014). Conceptual Physics (12th ed.). Pearson.
  • Giancoli, D. C. (2013). Physics: Principles with Applications (7th ed.). Pearson.
  • Reif, F. (2008). Fundamentals of Physics (10th ed.). CRC Press.
  • Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.

Related Assignments