91Ó°ÊÓ

In the exciting world of physics, not all collisions are created equal. An inelastic collision is one where the colliding bodies stick together after their encounter. This occurs when the kinetic energy isn't conserved due to factors like heat or deformation. However, a critical aspect of inelastic collisions is the conservation of momentum. This means the total momentum before the collision is equal to the total momentum after the collision.
Understanding this principle is key to solving problems involving situations where objects collide and move together post-impact. It enables us to find unknown variables, such as final velocity or the path angle of the colliding objects. In the case of our skater scenario, once they collide and hold on to each other, their combined mass moves off with a shared velocity and direction.

Momentum is a vector quantity, which means it has both magnitude and direction. It is expressed as the product of an object's mass and velocity. In scenarios involving two-dimensional motion, like our skaters, momentum can be broken into components. These components align with the standard horizontal (x-axis) and vertical (y-axis) directions.
The first skater has momentum only in the eastward (x) direction. Specifically, their momentum is calculated as the mass of the skater (50 kg) multiplied by their velocity (3.00 m/s), giving us 150 kg·m/s.
On the other hand, the second skater's momentum is directed south (y-axis). It is calculated by multiplying their mass (70 kg) by their speed (7.00 m/s), resulting in 490 kg·m/s. By breaking down each skater's momentum into components, we can easily manage the multi-dimensional analysis required for solving subsequent parts of the problem.

After the collision, when the skaters cling together and move as one, their combined mass makes it necessary to determine a single final velocity vector. This requires calculating both x and y components of this velocity. We utilize the conservation of momentum here:

• For the x-component: \[(m_1 + m_2) \cdot v_{f_x} = p_x = 150 \, \text{kg m/s} \]
• For the y-component: \[(m_1 + m_2) \cdot v_{f_y} = p_y = 490 \, \text{kg m/s} \]

Dividing the total momentum by the combined mass gives us:

• \[v_{f_x} = \frac{150}{120} = 1.25 \, \text{m/s} \]
• \[v_{f_y} = \frac{490}{120} \approx 4.08 \, \text{m/s} \]

The final velocity magnitude is obtained through the Pythagorean Theorem: \[v_f = \sqrt{(v_{f_x})^2 + (v_{f_y})^2} \approx 4.26 \, \text{m/s} \]. This calculation ensures we comprehend both the speed and the direction of the skaters after collision.

Once we've found the velocity components and hence the velocity magnitude, determining the direction is the next step. The question asks for the angle south of east, which is essentially measured from the positive x-axis towards the combined movement direction.
Using basic trigonometric principles, particularly the tangent function, we can solve for that angle:\[ \tan(\theta) = \frac{v_{f_y}}{v_{f_x}} = \frac{4.08}{1.25} \approx 3.264 \].
By calculating the inverse tangent, we get the angle:\[ \theta = \tan^{-1}(3.264) \approx 72.0^\circ \]. This angle gives a clear idea of the skaters' path, illustrating how they move on their new trajectory post-collision south of east. Understanding how to calculate such angles is crucial when considering vector directions in physics.