A quarterback takes the ball from the line of scrimmage, runs backwards for 5.0 yards, then runs sideways parallel to the line of scrimmage for 13.0 yards. At this point, he throws a 40.0 yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football's resultant displacement?
To find the magnitude of the football's resultant displacement, we can use the Pythagorean theorem. Let's break down the quarterback's movements into horizontal and vertical components:
1. The quarterback runs backwards for 5.0 yards, which is a negative movement in the horizontal direction.
2. The quarterback then runs sideways parallel to the line of scrimmage for 13.0 yards, which is a positive movement in the horizontal direction.
3. Finally, the quarterback throws a forward pass straight downfield for 40.0 yards. Since this is a perpendicular movement, it only contributes to the vertical component.
We can calculate the horizontal displacement by subtracting the negative movement from the positive movement:
Horizontal displacement = 13.0 yards - 5.0 yards = 8.0 yards
The vertical displacement is given by the forward pass, which is 40.0 yards.
Now, using the Pythagorean theorem, we can find the resultant displacement:
Resultant displacement = √(Horizontal displacement² + Vertical displacement²) = √(8.0² + 40.0²) yards
Calculating this value gives us:
Resultant displacement = √(64 + 1600) = √1664 yards ≈ 40.78 yards
Therefore, the magnitude of the football's resultant displacement is approximately 40.78 yards.