Final answer:
The displacement of the football from the line of scrimmage to the end zone is calculated by vectorial addition of the three segments of the quarterback's path. By breaking each run into horizontal and vertical components and summing them, we can determine the overall displacement magnitude and direction.
Step-by-step explanation:
To calculate the displacement of the football from the line of scrimmage to the end zone, where the quarterback scored, we will need to consider the individual segments of his run and combine them vectorially. The quarterback's displacement can be calculated step by step as follows:
- The quarterback first moves backwards for 12.0 yards.
- Next, he moves 5.7 yards at an angle of 52° north of west.
- Finally, he runs 38.8 yards at an angle of 12° east of north.
The overall displacement is the vector sum of these three movements. Each segment can be broken down into horizontal (x) and vertical (y) components using trigonometric functions (cosine for x, sine for y). The displacement for each segment, in yards, is as follows:
- First segment (backwards): dx=0, dy = -12.0
- Second segment (north of west): dx = 5.7*cos(128°), dy = 5.7*sin(128°)
- Third segment (east of north): dx = 38.8*sin(12°), dy = 38.8*cos(12°)
Sum the x and y components separately to find the total displacement components (Dx and Dy). Then, apply the Pythagorean theorem to calculate the magnitude of the displacement vector. The direction of the displacement can be found using the arctangent function (Tan-1) on the ratio of y-component to x-component.