Question
A football player runs directly down the field for 35 m before turning to
the right at an angle of 25° from his original direction and running an
additional 15 m before getting tackled. What is the magnitude and direc-
tion of the runner’s total displacement?
the right at an angle of 25° from his original direction and running an
additional 15 m before getting tackled. What is the magnitude and direc-
tion of the runner’s total displacement?
Answers
Answered by
GPT 3.5
To find the magnitude and direction of the runner's total displacement, we can break it down into horizontal and vertical components.
Horizontal component:
The 35 m run down the field is in the horizontal direction, so its horizontal component is 35 m.
Vertical component:
The runner turns to the right at an angle of 25°. We can consider this as a right triangle, where the 35 m side is the adjacent side and the unknown displacement is the hypotenuse. Using trigonometry, we can find the vertical component.
cos(25°) = adjacent / hypotenuse
cos(25°) = 35 m / hypotenuse
hypotenuse = 35 m / cos(25°)
hypotenuse ≈ 38.14 m
Now we can find the vertical component:
Vertical component = hypotenuse * sin(25°)
Vertical component = 38.14 m * sin(25°)
Vertical component ≈ 16.32 m
Total displacement:
To find the total displacement, we can use the Pythagorean theorem since the horizontal and vertical components are perpendicular to each other.
Total displacement = √(horizontal component^2 + vertical component^2)
Total displacement = √((35 m)^2 + (16.32 m)^2)
Total displacement ≈ √(1225 m^2 + 266.3424 m^2)
Total displacement ≈ √(1491.3424 m^2)
Total displacement ≈ 38.62 m
The magnitude of the runner's total displacement is approximately 38.62 m.
Direction:
To find the direction, we can use trigonometry again. The tangent of the angle is equal to the vertical component divided by the horizontal component.
tan(θ) = vertical component / horizontal component
tan(θ) = 16.32 m / 35 m
θ = arctan(16.32 m / 35 m)
θ ≈ 25.831°
The direction of the runner's total displacement is approximately 25.831° from his original direction.
Horizontal component:
The 35 m run down the field is in the horizontal direction, so its horizontal component is 35 m.
Vertical component:
The runner turns to the right at an angle of 25°. We can consider this as a right triangle, where the 35 m side is the adjacent side and the unknown displacement is the hypotenuse. Using trigonometry, we can find the vertical component.
cos(25°) = adjacent / hypotenuse
cos(25°) = 35 m / hypotenuse
hypotenuse = 35 m / cos(25°)
hypotenuse ≈ 38.14 m
Now we can find the vertical component:
Vertical component = hypotenuse * sin(25°)
Vertical component = 38.14 m * sin(25°)
Vertical component ≈ 16.32 m
Total displacement:
To find the total displacement, we can use the Pythagorean theorem since the horizontal and vertical components are perpendicular to each other.
Total displacement = √(horizontal component^2 + vertical component^2)
Total displacement = √((35 m)^2 + (16.32 m)^2)
Total displacement ≈ √(1225 m^2 + 266.3424 m^2)
Total displacement ≈ √(1491.3424 m^2)
Total displacement ≈ 38.62 m
The magnitude of the runner's total displacement is approximately 38.62 m.
Direction:
To find the direction, we can use trigonometry again. The tangent of the angle is equal to the vertical component divided by the horizontal component.
tan(θ) = vertical component / horizontal component
tan(θ) = 16.32 m / 35 m
θ = arctan(16.32 m / 35 m)
θ ≈ 25.831°
The direction of the runner's total displacement is approximately 25.831° from his original direction.