Question

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.08 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.

Answer 1

Answer:

The final speeds of the yellow and orange shuffleboard disks are 3.015 meters per second and 3.989 meters per second, respectively.

Explanation:

In this case, we see a two-dimension system formed by two shuffleboard disks, which do not experiment any effect from external forces, so that the Principle of Linear Momentum can be applied. The equations of equilibrium are respectively:

$m_{Y}\cdot \vec v_{Y,o}+m_{O}\cdot \vec v_{O,o} = m_{Y}\cdot \vec v_{Y}+m_{O}\cdot \vec v_{O}$ (Eq. 1)

Where:

$m_{Y}$, $m_{O}$ - Masses of the yellow and orange shuffleboard disks, measured in kilograms.

$\vec v_{Y,o}$, $\vec v_{O,o}$ - Initial vectors velocity of the yellow and orange shuffleboard disks, measured in meters per second.

$\vec v_{Y}$, $\vec v_{O}$ - Final vectors velocity of the yellow and orange shuffleboard disks, measured in meters per second.

If we know that $m_{Y} = m_{O}$, then the system is now reduced into this form:

$\vec v_{Y,o}+\vec v_{O,o} = \vec v_{Y}+\vec v_{O}$ (Eq. 2)

Given that $\vec v_{Y,o} = \left(0\,\frac{m}{s}, 0\,\frac{m}{s}\right)$, $\vec v_{O,o} = \left(5\,\frac{m}{s}, 0\,\frac{m}{s} \right)$, $\vec v_{Y} = v_{Y}\cdot \left(\cos 52.92^{\circ},-\sin 52.92^{\circ}\right)$ and $\vec v_{O} = v_{O}\cdot (\cos 37.08^{\circ},\sin 37.08^{\circ})$, then we find these two equations of equilibrium:

x-Direction:

$v_{Y}\cdot \cos 52.92^{\circ}+v_{O}\cdot \cos 37.08^{\circ} = 5\,\frac{m}{s}$ (Eq. 3)

y-Direction:

$-v_{Y}\cdot \sin 52.92^{\circ}+v_{O}\cdot \sin 37.08^{\circ} = 0\,\frac{m}{s}$ (Eq. 4)

The solution of this system of linear equations is:

$v_{Y} = 3.015\,\frac{m}{s}$ and $v_{O} = 3.989\,\frac{m}{s}$

The final speeds of the yellow and orange shuffleboard disks are 3.015 meters per second and 3.989 meters per second, respectively.