Mario, a hockey player, is skating due south at a speed of 7.05 m/s relative to the ice. A teammate passes the puck to him. The puck has a speed of 9.69 m/s and is moving in a direction of 16.2 ° west of south, relative to the ice. What are (a) the magnitude and (b) direction (relative to due south) of the puck's velocity, as observed by Mario?
Well, it seems like Mario really has his hands/paws full with this one! Let's break it down with some icy humor, shall we?
(a) The magnitude of the puck's velocity can be found by using some sweet trigonometry skills. We have the speeds of both Mario and the puck, so we can use the Pythagorean theorem to combine them. So, buckle up, because it's hockey math time!
Magnitude = sqrt(Mario's speed^2 + Puck's speed^2 - 2 * Mario's speed * Puck's speed * cos(16.2°))
Now, go grab a calculator and let's crunch those numbers!
(b) As for the direction, we're looking for the angle relative to due south. Since the puck is heading west of south, we just need to subtract the angle from 90 degrees to get the answer.
Direction = 90° - 16.2°
And voila! You've got the magnitude and direction of the puck's velocity as observed by Mario. Have fun juggling those icy numbers, my friend!
To solve this problem, we need to break down the velocities of Mario and the puck into their horizontal (x) and vertical (y) components. Then we can add the corresponding components to find the resultant velocity as observed by Mario.
Let's assume that the positive x-direction is due south, and the positive y-direction is due east (at a right angle to the x-direction).
Step 1: Find the x-component of Mario's velocity:
Since Mario is skating due south, there is no x-component to his velocity.
Step 2: Find the y-component of Mario's velocity:
Since Mario is skating due south, his y-velocity is zero.
Step 3: Convert the puck's magnitude and direction into its x and y-components.
The magnitude of the puck's velocity is 9.69 m/s. To find its x and y-components, we need to use trigonometry. The angle west of south is 16.2°.
The x-component of the puck's velocity, Vx, can be found by:
Vx = magnitude * cos(angle)
Vx = 9.69 m/s * cos(16.2°)
The y-component of the puck's velocity, Vy, can be found by:
Vy = magnitude * sin(angle)
Vy = 9.69 m/s * sin(16.2°)
Step 4: Find the resultant velocity of the puck as observed by Mario.
To find the resultant velocity, we need to add the x-components and y-components separately.
The x-component of the resultant velocity of the puck observed by Mario is the sum of the x-components of Mario's velocity and the puck's velocity: Vx_puck + Vx_Mario = 9.69 m/s * cos(16.2°).
The y-component of the resultant velocity of the puck observed by Mario is the sum of the y-components of Mario's velocity and the puck's velocity: Vy_puck + Vy_Mario = 9.69 m/s * sin(16.2°).
Step 5: Calculate the magnitude and direction of the resultant velocity.
The magnitude of the resultant velocity can be calculated using the Pythagorean theorem:
Resultant velocity = sqrt((Resultant Vx)^2 + (Resultant Vy)^2)
The direction of the resultant velocity relative to due south can be found using the inverse tangent function:
Direction = atan(Resultant Vy / Resultant Vx)
Now you can calculate the magnitude and direction of the puck's velocity as observed by Mario by plugging in the values calculated in steps 3 and 4 into the formulas above.