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.
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?
1 answer