A baseball is hit by Aaron Judge, a New York Yankees baseball team member, at an amazing velocity of 110 miles/hour at an angle of 35 degrees from the horizontal from home plate in Yankee Stadium in New York City. The baseball was initially launched from an elevation of 4 feet above ground. The distance from home plate to the outfield wall is 408 feet at center field. The height of the outfield wall is approximately 8 feet (It varies slightly across the outfield). You should first draw a two-dimensional picture of the baseball’s trajectory (x, y coordinate system) to assist in developing answers to the questions below. (a)a. Assuming no wind resistance, would the baseball go over outfield the wall for a home run? Would it land short of the wall? Would it hit the wall? Note: You must first convert all distances (in feet) from the British system into the metric system (in meters).
Vo = 110mi/h * 1600m/mi * 1h/3600s = 48.89m/s[35o]. = Initial velocity.
Xo = 48.89*Cos35 = 40.0 m/s. = Hor.
Yo = 48.89*sin35 = 28.0 m/s. = Ver.
4ft * 1m/3.3ft = 1.21 m.
8ft = 2.42 m.
408ft * 1m/3.3ft = 123.6 m.
Y = Yo + g+Tr = 0.
28 - 9.8Tr = 0
Tr = 2.86 s. = Rise time or time to reach max ht.
hmax = ho + Yo*Tr + 0.5g*(Tr)^2.
hmax = 1.21 + 28*2.86 - 4.9*(2.86)^2 = 41.2 m. = Max ht.
d = 0.5g*(Tf)^2 = 41.2-2.42.
4.9(Tf)^2 = 38.79
Tf = 2.81 s. = Fall time from max ht. to top of fence.
Range = Xo*(Tr+Tf) = 40*(2.86+2.81) = 218.8 m.
The range is greater than the distance from home plate to the center field wall(123.6m). Therefore, it's a home run!!
To determine whether the baseball will go over the outfield wall for a home run, we need to analyze its trajectory. Let's break it down step by step:
1. Convert distances from feet to meters:
- Distance to outfield wall: 408 feet = 124.3896 meters
- Elevation from home plate to launch point: 4 feet = 1.2192 meters
- Height of the outfield wall: 8 feet = 2.4384 meters
2. Draw a two-dimensional diagram:
- Create an x-y coordinate system where the x-axis represents the horizontal distance and the y-axis represents the vertical distance.
- Place the launch point at coordinates (0, 1.2192) meters, and the center of the outfield wall at coordinates (124.3896, 2.4384) meters.
3. Analyze the trajectory:
- The baseball was launched at an angle of 35 degrees from the horizontal.
- The initial velocity of the baseball is given as 110 miles/hour. To use this information, convert the velocity to meters/second:
- 110 miles/hour = 49.176 meters/second
4. Split the initial velocity into horizontal and vertical components:
- Horizontal component: v_x = velocity * cos(angle) = 49.176 * cos(35) meters/second
- Vertical component: v_y = velocity * sin(angle) = 49.176 * sin(35) meters/second
5. Calculate the time of flight:
- We can determine the time of flight by considering the vertical motion of the baseball.
- The equation to calculate time is t = (2 * v_y) / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Substitute the values to find t.
6. Calculate the horizontal distance traveled:
- The horizontal distance traveled, known as the range (R), can be calculated using the equation: R = v_x * t.
7. Determine whether the baseball clears the outfield wall:
- Compare the horizontal distance traveled to the distance to the outfield wall.
- If R > 124.3896 meters, the ball goes over the outfield wall for a home run.
- If R = 124.3896 meters, the ball hits the wall.
- If R < 124.3896 meters, the ball lands short of the wall.
By following these steps and applying the appropriate calculations, we can determine whether the baseball hit by Aaron Judge goes over the outfield wall, hits the wall, or lands short of the wall, assuming no wind resistance.