In a World Cup soccer match, Juan is running due north toward the goal with a speed of 7.70 relative to the ground. A teammate passes the ball to him. The ball has a speed of 13.3 and is moving in a direction of 30.5 east of north, relative to the ground.
a) what is the magnitude of the ball's velocity relative to Juan?
b) what is the direction of the ball's velocity relative to Juan?
To solve this problem, we can break down the components of the ball's velocity relative to Juan into horizontal and vertical components.
a) To find the magnitude of the ball's velocity relative to Juan, we can use the Pythagorean theorem:
Magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
The ball's horizontal component of velocity can be found using trigonometry:
horizontal component = ball speed * cos(angle)
Substituting the given values, we get:
horizontal component = 13.3 * cos(30.5)
Next, we can find the ball's vertical component of velocity:
vertical component = ball speed * sin(angle)
Substituting the given values, we get:
vertical component = 13.3 * sin(30.5)
Finally, we can calculate the magnitude:
Magnitude = sqrt((13.3 * cos(30.5))^2 + (13.3 * sin(30.5))^2)
Using a calculator, the magnitude comes out to be approximately 11.81 m/s.
Therefore, the magnitude of the ball's velocity relative to Juan is 11.81 m/s.
b) To find the direction of the ball's velocity relative to Juan, we can use trigonometry again:
direction = arctan(vertical component / horizontal component)
Substituting the values, we get:
direction = arctan((13.3 * sin(30.5)) / (13.3 * cos(30.5)))
Using a calculator, the direction comes out to be approximately 30.5 degrees east of north.
Therefore, the direction of the ball's velocity relative to Juan is 30.5 degrees east of north.
To find the magnitude of the ball's velocity relative to Juan (a), we can use vector addition.
Step 1: Convert the velocities into vector form.
Juan's velocity: 7.70 m/s due north
Ball's velocity: 13.3 m/s at 30.5° east of north
Step 2: Break down the ball's velocity into its northward and eastward components.
The northward component: 13.3 m/s * sin(30.5°)
The eastward component: 13.3 m/s * cos(30.5°)
Step 3: Add the components of the ball's velocity to Juan's velocity.
Magnitude of the ball's velocity relative to Juan: √[(7.70 m/s + eastward component)^2 + (7.70 m/s + northward component)^2]
Now, let's calculate the values.
Step 1: Convert the velocities into vector form.
Juan's velocity: (0 m/s, 7.70 m/s)
Ball's velocity: (13.3 m/s * cos(30.5°), 13.3 m/s * sin(30.5°))
Step 2: Break down the ball's velocity into its northward and eastward components.
The northward component: 13.3 m/s * sin(30.5°) ≈ 6.825 m/s
The eastward component: 13.3 m/s * cos(30.5°) ≈ 11.423 m/s
Step 3: Add the components of the ball's velocity to Juan's velocity.
Magnitude of the ball's velocity relative to Juan: √[(0 m/s + 11.423 m/s)^2 + (7.70 m/s + 6.825 m/s)^2]
Now, let's calculate the magnitude of the ball's velocity relative to Juan.
Magnitude = √[(11.423 m/s)^2 + (14.525 m/s)^2] ≈ 18.52 m/s
So, the magnitude of the ball's velocity relative to Juan is approximately 18.52 m/s.
To find the direction of the ball's velocity relative to Juan (b), we can use the inverse tangent function.
Angle (θ) = arctan((northward component)/(eastward component))
Angle (θ) = arctan((6.825 m/s)/(11.423 m/s))
Now, let's calculate the angle.
Angle (θ) ≈ 29.4°
So, the direction of the ball's velocity relative to Juan is approximately 29.4° east of north.