A 89.0 kg fullback running east with a speed of 5.00 m/s is tackled by a 97.0 kg opponent running north with a speed of 6.00 m/s. (a) Calculate the velocity of the players immediately after the tackle.
To calculate the velocity of the players immediately after the tackle, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass * velocity
Let's assume that the positive x-direction is east and the positive y-direction is north.
The momentum before the collision can be calculated as follows:
Momentum of the fullback in the x-direction (Px_fullback) = mass_fullback * velocity_fullback
Px_fullback = 89.0 kg * 5.00 m/s
Momentum of the opponent in the y-direction (Py_opponent) = mass_opponent * velocity_opponent
Py_opponent = 97.0 kg * 6.00 m/s
Since these two momenta are in perpendicular directions, we can write:
Total momentum before the collision (P_initial) = sqrt(Px_fullback^2 + Py_opponent^2)
Now, according to the conservation of momentum, the total momentum after the collision (P_final) should be equal to P_initial.
The total momentum after the collision can be calculated as follows:
Momentum of the fullback after the collision in the x-direction (Px_fullback_final) = mass_fullback * velocity_fullback_final
Momentum of the opponent after the collision in the y-direction (Py_opponent_final) = mass_opponent * velocity_opponent_final
Again, these two momenta are perpendicular to each other, so we can write:
Total momentum after the collision (P_final) = sqrt(Px_fullback_final^2 + Py_opponent_final^2)
Since the total momentum before the collision (P_initial) is equal to the total momentum after the collision (P_final), we can equate these two equations:
sqrt(Px_fullback^2 + Py_opponent^2) = sqrt(Px_fullback_final^2 + Py_opponent_final^2)
Now we have an equation with two unknowns (Px_fullback_final and Py_opponent_final). We need one more equation to solve for these unknowns.
We can use the conservation of kinetic energy to find the second equation. According to the conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy of an object is given by the product of one-half times the mass times the square of the velocity:
Kinetic energy = (1/2) * mass * velocity^2
The total kinetic energy before the collision can be calculated as follows:
Kinetic energy of the fullback before the collision (KE_fullback_initial) = (1/2) * mass_fullback * velocity_fullback^2
Kinetic energy of the opponent before the collision (KE_opponent_initial) = (1/2) * mass_opponent * velocity_opponent^2
The total kinetic energy before the collision (KE_initial) is equal to the sum of these two energies:
KE_initial = KE_fullback_initial + KE_opponent_initial
The total kinetic energy after the collision can be calculated as follows:
Kinetic energy of the fullback after the collision (KE_fullback_final) = (1/2) * mass_fullback * velocity_fullback_final^2
Kinetic energy of the opponent after the collision (KE_opponent_final) = (1/2) * mass_opponent * velocity_opponent_final^2
Again, the total kinetic energy after the collision (KE_final) is equal to the sum of these two energies:
KE_final = KE_fullback_final + KE_opponent_final
Since the total kinetic energy before the collision (KE_initial) is equal to the total kinetic energy after the collision (KE_final), we can equate these two equations:
KE_fullback_initial + KE_opponent_initial = KE_fullback_final + KE_opponent_final
Now we have two equations:
(1) sqrt(Px_fullback^2 + Py_opponent^2) = sqrt(Px_fullback_final^2 + Py_opponent_final^2)
(2) KE_fullback_initial + KE_opponent_initial = KE_fullback_final + KE_opponent_final
By solving these two equations simultaneously, we can find the values of Px_fullback_final and Py_opponent_final, which represents the velocities of the fullback and the opponent immediately after the tackle.