Mario, a hockey player, is skating due south at a speed of 5.4 m/s relative to the ice. A teammate passes the puck to him. The puck has a speed of 10.0 m/s and is moving in a direction of 27° west of south, relative to the ice. What are the magnitude and direction (relative to due south) of the puck's velocity, as observed by Mario?

Question

Mario, a hockey player, is skating due south at a speed of 5.4 m/s relative to the ice. A teammate passes the puck to him. The puck has a speed of 10.0 m/s and is moving in a direction of 27° west of south, relative to the ice. What are the magnitude and direction (relative to due south) of the puck's velocity, as observed by Mario?

Answer 1

To determine the magnitude and direction of the puck's velocity as observed by Mario, we can analyze the horizontal and vertical components separately.

First, let's find the horizontal component. Since Mario is skating due south, there is no horizontal motion in this direction. So, the horizontal component of the puck's velocity, as observed by Mario, is 0 m/s.

Next, let's find the vertical component. We can use trigonometry to find the vertical component of the puck's velocity. The magnitude of the puck's velocity relative to the ice is given as 10.0 m/s, and the direction is 27° west of south. To find the vertical component, we need to calculate the sine of the angle:

sin(27°) = vertical component / 10.0 m/s

vertical component = sin(27°) * 10.0 m/s

To find the magnitude of the puck's velocity as observed by Mario, we use the Pythagorean theorem:

magnitude = sqrt(horizontal component^2 + vertical component^2)

magnitude = sqrt(0^2 + (sin(27°) * 10.0)^2)

Finally, we can calculate the direction of the puck's velocity relative to due south. Since the horizontal component is 0 m/s, the direction will be the same as the direction of the vertical component. We can find this angle using inverse trigonometric functions:

angle = arctan(vertical component / horizontal component)

angle = arctan((sin(27°) * 10.0) / 0)

The magnitude of the puck's velocity, as observed by Mario, is the value obtained from the magnitude calculation, and the direction is the value obtained from the angle calculation.

Please note that if you need the numerical values of the magnitude and direction of the puck's velocity observed by Mario, you need to substitute the appropriate values into the above formulas and perform the calculations.