A 70.0 kg ice hockey goalie, originally at rest, has a 0.110 kg hockey puck slapped at him at a velocity of 37.5 m/s. Suppose the goalie and the puck have a perfectly elastic collision, such that the puck is reflected back in the direction from which it came.

What would the final velocities v_goalie and v_puck of the goalie and the puck, respectively, be in this case?

To solve this problem, we need to use the principles of **conservation of momentum** and **conservation of kinetic energy**, which apply in a perfectly elastic collision.

### Given dataTo solve this problem, we need to use the principles of **conservation of momentum** and **conservation of kinetic energy**, which apply in a perfectly elastic collision.

### Given 
- Mass of the goalie (\( m_{\text{goalie}} \)) = 70.0 kg
- Initial velocity of the goalie (\( u_{\text{goalie}} \)) = 0 m/s
- Mass of the puck (\( m_{\text{puck}} \)) = 0.110 kg
- Initial velocity of the puck (\( u_{\text{puck}} \)) = 37.5 m/s (toward the goalie)

Let:
- Final velocity of the goalie = \( v_{\text{goalie}} \)
- Final velocity of the puck = \( v_{\text{puck}} \)

Since the collision is perfectly elastic, both **momentum** and **kinetic energy** are conserved.

### Step 1: Conservation of Momentum
The total momentum before the collision is equal to the total momentum after the collision.

\[
m_{\text{goalie}} \cdot u_{\text{goalie}} + m_{\text{puck}} \cdot u_{\text{puck}} = m_{\text{goalie}} \cdot v_{\text{goalie}} + m_{\text{puck}} \cdot v_{\text{puck}}
\]

Substituting the given values:

\[
(70.0 \, \text{kg}) \cdot 0 + (0.110 \, \text{kg}) \cdot 37.5 = (70.0 \, \text{kg}) \cdot v_{\text{goalie}} + (0.110 \, \text{kg}) \cdot v_{\text{puck}}
\]

Simplifying,

\[
4.125 = 70.0 \cdot v_{\text{goalie}} + 0.110 \cdot v_{\text{puck}}
\]

### Step 2: Conservation of Kinetic Energy
The total kinetic energy before the collision is equal to the total kinetic energy after the collision.

\[
\frac{1}{2} m_{\text{goalie}} u_{\text{goalie}}^2 + \frac{1}{2} m_{\text{puck}} u_{\text{puck}}^2 = \frac{1}{2} m_{\text{goalie}} v_{\text{goalie}}^2 + \frac{1}{2} m_{\text{puck}} v_{\text{puck}}^2
\]

Substituting the given values:

\[
\frac{1}{2} \cdot (70.0) \cdot (0)^2 + \frac{1}{2} \cdot (0.110) \cdot (37.5)^2 = \frac{1}{2} \cdot (70.0) \cdot v_{\text{goalie}}^2 + \frac{1}{2} \cdot (0.110) \cdot v_{\text{puck}}^2
\]

Calculating the initial kinetic energy of the puck:

\[
0 + \frac{1}{2} \cdot 0.110 \cdot 37.5^2 = \frac{1}{2} \cdot 70.0 \cdot v_{\text{goalie}}^2 + \frac{1}{2} \cdot 0.110 \cdot v_{\text{puck}}^2
\]

\[
77.34375 = 35.0 \cdot v_{\text{goalie}}^2 + 0.055 \cdot v_{\text{puck}}^2
\]

### Step 3: Using Relative Velocity in Elastic Collisions
For a perfectly elastic collision, the relative velocity of separation equals the relative velocity of approach. This can be formulated as:

\[
u_{\text{puck}} - u_{\text{goalie}} = -(v_{\text{puck}} - v_{\text{goalie}})
\]

Since \( u_{\text{goalie}} = 0 \):

\[
37.5 = -(v_{\text{puck}} - v_{\text{goalie}})
\]

So,

\[
v_{\text{puck}} = v_{\text{goalie}} - 37.5
\]

### Step 4: Solve the System of Equations
Now we have two equations:

1. \( 4.125 = 70.0 \cdot v_{\text{goalie}} + 0.110 \cdot v_{\text{puck}} \)
2. \( v_{\text{puck}} = v_{\text{goalie}} - 37.5 \)

Substitute \( v_{\text{puck}} = v_{\text{goalie}} - 37.5 \) into the first equation:

\[
4.125 = 70.0 \cdot v_{\text{goalie}} + 0.110 \cdot (v_{\text{goalie}} - 37.5)
\]

Expanding and solving for \( v_{\text{goalie}} \):

\[
4.125 = 70.0 \cdot v_{\text{goalie}} + 0.110 \cdot v_{\text{goalie}} - 4.125
\]

\[
8.25 = 70.11 \cdot v

A 70.0 kg ice hockey goalie, originally at rest, has a 0.110 kg hockey puck slapped at him at a velocity of 37.5 m/s. Suppose the goalie and the puck have a perfectly elastic collision, such that the puck is reflected back in the direction from which it came. What would the final velocities v_goalie and v_puck of the goalie and the puck, respectively, be in this case?

Calculate the final velocities of a 70.0 kg ice hockey goalie and a 0.110 kg puck in a perfectly elastic collision with initial puck speed of 37.5 m/s. Explore key physics concepts such as conservation of momentum and kinetic energy, with step-by-step solutions for high school students in Grades 11-12.

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