A hockey puck with mass \(0.250 \mathrm{~kg}\) traveling along the blue line (a blue-colored straight line on the ice in a hockey rink) at \(1.50 \mathrm{~m} / \mathrm{s}\) strikes a stationary puck with the same mass. The first puck exits the collision in a direction that is \(30.0^{\circ}\) away from the blue line at a speed of \(0.750 \mathrm{~m} / \mathrm{s}\) (see the figure). What is the direction and magnitude of the velocity of the second puck after the collision? Is this an elastic collision?

In the fascinating world of physics, conservation of momentum is among the most steadfast principles, providing insights into interactions like those seen in the game of hockey. Imagine two hockey pucks sliding across the ice; when they collide, the total momentum of this system before and after the collision remains constant (if no external forces act on it). The concept is encapsulated in the elegant equation:

\( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \),

where \(m_1\) and \(m_2\) are the masses of the pucks, and \(v_{1i}, v_{2i}\) and \(v_{1f}, v_{2f}\) are their initial and final velocities, respectively. In our example, the momenta of the two pucks before and after they crash on the rink must be equal when you combine their individual momenta. The equation comes into play for both the horizontal (x) and vertical (y) components, reinforcing that momentum is a vector quantity comprising both magnitude and direction. By solving the momentum equations for each component separately, we can find the final velocities post-collision, which leads to a comprehensible illustration of the principle in action.

Now let's glide into the realm of kinetic energy (KE), a term that essentially quantifies the energy of motion. For a single puck coasting on the ice, the kinetic energy can be calculated using the equation:

\( KE = \frac{1}{2}mv^2 \),

where \(m\) represents the mass and \(v\) is the velocity. It’s a scalar quantity so it doesn’t have direction but is tied intrinsically to how fast the puck is moving and how much it weighs. In our scenario, to verify whether the collision is elastic, meaning no kinetic energy is lost, we compare the total kinetic energy before and after the pucks collide. If the sum of the pucks' kinetic energies remains unchanged post-collision, as it does in this case (

• Initial KE: \( 0.28125 J \)
• Final KE: \( 0.28125 J \)

), we can affirm the collision is indeed elastic. An elastic collision is akin to a perfectly choreographed dance—diverse steps and movements but the dance's energy remains constant.

Navigating into the territory of vector components requires an understanding of both magnitude and direction. Vectors are fundamental in physics as they represent quantities with both of these attributes, like velocity.

Think of the ice hockey puck not just moving in a straight line but perhaps in two dimensions, slicing across the ice at an angle. To address such a scenario, one would decompose the vector, in this case, the puck’s velocity, into both x (horizontal) and y (vertical) components. This can be done using trigonometry, specifically,

• for the x-component: \( v_x = v \cos(\theta) \)
• for the y-component: \( v_y = v \sin(\theta) \)

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where \(\theta\) is the angle with respect to the horizontal. Once the x and y components are known, they can be aggregated to understand the overall velocity and direction of the motion. The final velocity can be found using the Pythagorean theorem: \( v = \sqrt{v_x^2 + v_y^2} \) and the angle by the inverse tangent function: \( \theta = \arctan\left(\frac{|v_y|}{|v_x|}\right) \). In regards to the hockey pucks, dissecting the velocity into its vector components allows us to evaluate the outcome of the collision with greater accuracy.