Problem 87 - College Physics

Question

A young hockey player with a mass of \(30.0 \mathrm{~kg}\) is initially moving at \(2.00 \mathrm{~m} / \mathrm{s}\) to the east. He intercepts and catches on the stick a puck initially moving at \(35.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(\theta=60^{\circ}\) (vFig. 6.41). Assume that the puck's mass is \(0.180 \mathrm{~kg}\) and the player and puck form a single object for a few seconds. (a) Determine the direction angle and speed of the puck and skater after the collision. (b) Was this collision elastic or inelastic? Prove your answer with numbers.

Step-by-Step Solution

Answer

The final velocity magnitude is approximately 3.76 m/s at 10.8° east of north. The collision is inelastic.

Step 1: Identify Given Data

The problem provides us with the mass and velocity of both the player and the puck. Player mass, \( m_{p} = 30.0 \; \text{kg} \) and velocity, \( v_{p} = 2.00 \; \text{m/s} \) east. Puck mass, \( m_{k} = 0.180 \; \text{kg} \) and velocity, \( v_{k} = 35.0 \; \text{m/s} \) at \( \theta = 60^{\circ} \).

Step 2: Apply Conservation of Momentum

Momentum conservation is used to solve elastic or inelastic collisions. We separate the momentum into components: x (east) and y (north).\[\begin{aligned}&m_{p} v_{p} + m_{k} v_{kx} = (m_{p} + m_{k}) v \cos \phi, \&0 + m_{k} v_{ky} = (m_{p} + m_{k}) v \sin \phi,\end{aligned}\]where \( v_{kx} = v_{k}\cos(\theta) \) and \( v_{ky} = v_{k}\sin(\theta) \).

Step 3: Solve for Final Velocity Components

Substitute \( v_{kx} = 35.0 \cdot \cos(60^{\circ}) \) and \( v_{ky} = 35.0 \cdot \sin(60^{\circ}) \) into the equations. Calculate \[v_x = \frac{m_{p} v_{p} + m_{k} v_{k} \cos(\theta)}{m_{p} + m_{k}}\] and \[v_y = \frac{m_{k} v_{k} \sin(\theta)}{m_{p} + m_{k}}\].

Step 4: Calculate Magnitude and Direction of Final Velocity

The magnitude \( v \) of the final velocity is \[v = \sqrt{v_x^2 + v_y^2}\].The direction angle \( \phi \) is \[\phi = \tan^{-1}\left(\frac{v_y}{v_x}\right)\].

Step 5: Check Elasticity of Collision

Calculate the initial kinetic energy (KE) of both player and puck and the final KE of the combined mass. \[\begin{aligned}&\text{Initial KE} = \frac{1}{2}m_{p}v_{p}^2 + \frac{1}{2}m_{k}v_{k}^2, \&\text{Final KE} = \frac{1}{2}(m_{p} + m_{k})v^2.\end{aligned}\]Compare the initial and final kinetic energies to determine if the collision is elastic or inelastic (energy is not conserved in inelastic collisions).

Key Concepts

Inelastic Collisions

Inelastic collisions are events where two colliding objects stick together after impact or when kinetic energy is not conserved. This differs from elastic collisions, where both momentum and kinetic energy remain constant. In inelastic collisions, only momentum is conserved. The exchanged energy can be dissipated as heat or sound.

In our exercise, the young hockey player catches the puck on his stick. During this interaction, they move together momentarily. This means the collision is inelastic. When checking whether the collision is elastic or inelastic, compare the kinetic energy before and after the event.

• An elastic collision means the kinetic energy stays the same both before and after the collision.
• An inelastic collision results if the kinetic energy changes.

After performing calculations, you will often find that some kinetic energy is "lost" in inelastic collisions. This information helps confirm the nature of the collision.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. Its formula is \[KE = \frac{1}{2}mv^2\]where \(m\) is the mass and \(v\) is the velocity of the object. During collisions, especially in physics problems, we check if total kinetic energy is conserved to distinguish between elastic and inelastic collisions.

In the given problem, before the collision, each object has its own kinetic energy. The young player and the puck both contribute separately to the initial total kinetic energy:

• Initial kinetic energy of the hockey player is calculated using their mass and velocity.
• The puck's kinetic energy is considered with its respective mass and velocity—a bit trickier due to its angle.

After the collision, both objects combined move with a new velocity.
Comparing total initial kinetic energy with total final kinetic energy reveals energy conservation status. In our task, if the kinetic energy post-collision is lesser, it signals an inelastic interaction.

Vector Components

When dealing with forces, velocities or momenta that occur at angles, we often break them down into vector components to simplify calculations. These components typically communicate the x-direction and y-direction elements. Using trigonometry helps in determining these components:

Given a velocity \(v\) at an angle \(\theta\), the components can be calculated as:

• \(v_x = v\cos(\theta)\)
• \(v_y = v\sin(\theta)\)

In this problem, the puck's movement is at an angle to east, leading us to break it into two parts:

• The east (x-component) as \(v_{kx}\)
• The north (y-component) as \(v_{ky}\)

By analyzing each component separately, we can accurately apply conservation laws, like momentum, and find results such as direction and speed of post-collision movement.

Vector components simplify the handling of multiple forces or velocities acting simultaneously. Thus, they show up frequently in physics problems, allowing easier addition and subtraction in calculations.