Answer:
The second puck moves with a speed of 3.867 m/s due east and the first puck moves with a speed of 1.933 m/s, but in the Western direction after collision.
Explanation:
Let the mass of the first hockey puck = m₁ = 0.45 kg
Mass of the resting puck = m₂ = 0.90 kg
Velocity of the first hockey puck = v₁ = 5.80î m/s; magnitude = 5.80 m/s
Velocity of resting puck = v₂ = 0 m/s
Mass of the two pucks are the same after collision.
Velocity of first puck after collision = v₃ = ?
Velocity of second puck after collision = v₄ = ?
In an elastic collision, both momentum and kinetic energy are conserved.
Momentum before collision = momentum after collision
Momentum before collision
m₁v₁ + m₂v₂ = (0.45 × 5.8) + (0.9 × 0) = 2.61 kgm/s
Momentum after collision
m₁v₃ + m₂v₄ = 0.45v₃ + 0.9v₄
So, according to the law of conservation of Momentum,
2.61 = 0.45v₃ + 0.9v₄ (eqn 1)
Kinetic energy before collision = kinetic energy after collision
Kinetic energy before collision = (m₁v₁²/2) + (m₂v₂²/2) = (0.45 × 5.8²/2) + (0.9 × 0²/2) = 7.569 J
Kinetic energy after collision = (m₁v₃²/2) + (m₂v₄²/2) = (0.45v₃²/2) + (0.9v₄²/2) = 0.225v₃² + 0.45v₄²
Since kinetic energy is conserved,
7.569 = 0.225v₃² + 0.45v₄² (eqn 2)
2.61 = 0.45v₃ + 0.9v₄ (eqn 1)
7.569 = 0.225v₃² + 0.45v₄² (eqn 2)
Make v₃ the subject of formula from eqn 1,
v₃ = (2.61 - 0.9v₄)/0.45 = (5.8 - 2v₄)
Substitute the value of v₃ into eqn 2
7.569 = 0.225(5.8 - 2v₄)² + 0.45v₄²
7.569 = 0.225(33.64 - 23.2v₄ + 4v₄²) + 0.45v₄²
7.569 = 7.569 - 5.22v₄ + 0.9v₄² + 0.45v₄²
1.35v₄² - 5.22v₄ = 0
v₄² - 3.867v₄ = 0
v₄ (v₄ - 3.867) = 0
v₄ = 0 m/s or v₄ = 3.867 m/s
If v₄ = 0 m/s,
v₃ = (5.8 - 2v₄) = [5.8 - 2(0)] = 5.8 m/s
If v₄ = 3.867 m/s
v₃ = (5.8 - 2v₄) = [5.8 - 2(3.867)] = - 1.933 m/s
The (v₃,v₄) answers are (5.8,0) and (-1.933,3.867). Since the collision was stated to be elastic, the realistic answer is the (-1.933, 3.867) answer.
So, the answer is interpreted as,
The second puck moves with a speed of 3.867 m/s due east and the first puck moves with a speed of 1.933 m/s, but in the Western direction after collision.
