Ice hockey star Wayne gretzky is skatin at 13.0 m/s^1 towards a defender, who is in turn skating at 5.0 m/s^-1 toward gretzky.gretzky's weight is 756 N, that of the defender is 900 N. Immediately after the collision. gretzky is moving at 3.00 m/s^-1 in his original direction. Neglect exernal horizonal forces applied by the ice to the skaters during the collision.
A) What is the velocity of the defender immediately after the collision?
B) Calculate the change in total kinetic energy of the two players?
To find the answers, let's break down the problem into steps:
Step 1: Analyze the initial velocities
- The initial velocity of Wayne Gretzky (g) is 13.0 m/s in his original direction.
- The initial velocity of the defender (d) is 5.0 m/s towards Gretzky.
Step 2: Analyze the final velocities
- After the collision, Gretzky's final velocity is 3.00 m/s in his original direction.
- We need to find the final velocity of the defender (d').
Step 3: Apply conservation of momentum
- In a collision, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces.
- Mathematically, conservation of momentum can be expressed as: m1*v1 + m2*v2 = m1*v1' + m2*v2', where m1 and m2 are the masses of the objects, and v1, v2, v1', v2' are the corresponding velocities before and after the collision.
Step 4: Calculate the final velocity of the defender (d')
- Let's assume the mass of Gretzky (m1) is the same as the mass of the defender (m2) for simplicity.
- Using the conservation of momentum equation from Step 3, we can write it as: (m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
- Substituting the given values, we get: (m1 * 13.0) + (m2 * 5.0) = (m1 * 3.00) + (m2 * v2')
- The mass of Gretzky (m1) and the mass of the defender (m2) cancel out since they are equal.
- Simplifying the equation, we have: 13.0 + 5.0 = 3.00 + v2'
- Rearranging and solving for v2', we find: v2' = 13.0 + 5.0 - 3.00 = 15.0 m/s
- Therefore, the velocity of the defender immediately after the collision is 15.0 m/s in his original direction (towards Gretzky).
Step 5: Calculate the change in total kinetic energy of the two players
- The total kinetic energy before the collision is equal to the total kinetic energy after the collision, assuming energy losses due to non-elastic collision are negligible.
- The formula for kinetic energy is KE = (1/2) * mass * velocity^2.
- Let's calculate the initial kinetic energy (KE_initial) and final kinetic energy (KE_final) for both players.
For Gretzky:
- KE_initial (g) = (1/2) * mass * velocity^2 = (1/2) * 756 * 13.0^2 = 63708 J
- KE_final (g) = (1/2) * mass * velocity^2 = (1/2) * 756 * 3.00^2 = 3402.0 J
For the defender:
- KE_initial (d) = (1/2) * mass * velocity^2 = (1/2) * 900 * 5.0^2 = 11250 J
- KE_final (d') = (1/2) * mass * velocity^2 = (1/2) * 900 * 15.0^2 = 101250 J
- The change in total kinetic energy (ΔKE) of the two players is the difference between the total initial kinetic energy (KE_initial_total) and the total final kinetic energy (KE_final_total).
- ΔKE = KE_initial_total - KE_final_total = (KE_initial (g) + KE_initial (d)) - (KE_final (g) + KE_final (d')) = (63708 + 11250) - (3402.0 + 101250) = -35394 J
- The negative sign indicates that the total kinetic energy decreases after the collision.
Therefore, to answer the questions:
A) The velocity of the defender immediately after the collision is 15.0 m/s towards Gretzky.
B) The change in total kinetic energy of the two players is -35394 J.
Let's apply the principle of conservation of linear momentum to solve this problem.
According to the principle of conservation of linear momentum, the total momentum before the collision should be equal to the total momentum after the collision.
Step 1: Calculate the initial momentum of Gretzky and the defender.
Momentum is given by the formula: momentum = mass * velocity.
The mass of Gretzky = weight / acceleration due to gravity = 756 N / 9.8 m/s^2 = 77.14 kg
The initial momentum of Gretzky = mass of Gretzky * velocity of Gretzky
= 77.14 kg * 13.0 m/s
= 1003.82 kg*m/s
The mass of the defender = weight / acceleration due to gravity = 900 N / 9.8 m/s^2 = 91.84 kg
The initial momentum of the defender = mass of the defender * velocity of the defender
= 91.84 kg * (-5.0 m/s) (negative because the defender is moving in the opposite direction)
= -459.2 kg*m/s (negative because the direction is opposite)
Step 2: After the collision, Gretzky's velocity is given as 3.00 m/s in his original direction.
To find the velocity of the defender after the collision, we subtract the final momentum of Gretzky from the total initial momentum.
Final momentum of Gretzky = mass of Gretzky * final velocity of Gretzky
= 77.14 kg * 3.00 m/s
= 231.42 kg*m/s
Final momentum of the defender = Total initial momentum - Final momentum of Gretzky
= (1003.82 kg*m/s + (-459.2 kg*m/s)) - 231.42 kg*m/s
= 313.2 kg*m/s
Step 3: Calculate the velocity of the defender after the collision.
The velocity of the defender after the collision = Final momentum of the defender / mass of the defender
= 313.2 kg*m/s / 91.84 kg
= 3.41 m/s (rounded to two decimal places)
Therefore, the velocity of the defender immediately after the collision is 3.41 m/s.
Step 4: Calculate the change in total kinetic energy of the two players.
Change in total kinetic energy = Final total kinetic energy - Initial total kinetic energy
The initial total kinetic energy = 1/2 * mass of Gretzky * (velocity of Gretzky)^2 + 1/2 * mass of the defender * (velocity of the defender)^2
= 1/2 * 77.14 kg * (13.0 m/s)^2 + 1/2 * 91.84 kg * (-5.0 m/s)^2
= 6547.1 J
The final total kinetic energy = 1/2 * mass of Gretzky * (final velocity of Gretzky)^2 + 1/2 * mass of the defender * (final velocity of the defender)^2
= 1/2 * 77.14 kg * (3.0 m/s)^2 + 1/2 * 91.84 kg * (3.41 m/s)^2
= 1180.49 J
Change in total kinetic energy = 1180.49 J - 6547.1 J
= -5366.61 J (negative because the final kinetic energy is less than the initial)
Therefore, the change in total kinetic energy of the two players is -5366.61 J.