The velocity of the center of mass is (4.26 i + 8.11 j) m/s, and the total momentum of the system is (6.62 i + 1.74 j) kg*m/s.
a) To find the velocity of the center of mass, we need to use the formula:
Center of mass velocity = (m1 * v1 + m2 * v2) / (m1 + m2)
where m1 and m2 are the masses of the particles, and v1 and v2 are their respective velocities.
Given that the first particle has a mass of 2.09 kg and a velocity of (2.03 i - 2.92 j) m/s, and the second particle has a mass of 2.97 kg and a velocity of (0.99 i + 5.94 j) m/s, we can substitute these values into the formula:
Center of mass velocity = (2.09 kg * (2.03 i - 2.92 j) m/s + 2.97 kg * (0.99 i + 5.94 j) m/s) / (2.09 kg + 2.97 kg)
Simplifying the calculation, we get:
Center of mass velocity = (4.2577 i + 8.1126 j) m/s
Therefore, the velocity of the center of mass is (4.26 i + 8.11 j) m/s.
b) To find the total momentum of the system, we need to calculate the momentum of each particle and add them together.
The momentum of a particle is given by the formula:
Momentum = mass * velocity
For the first particle, with a mass of 2.09 kg and a velocity of (2.03 i - 2.92 j) m/s, the momentum is:
Momentum1 = 2.09 kg * (2.03 i - 2.92 j) m/s
For the second particle, with a mass of 2.97 kg and a velocity of (0.99 i + 5.94 j) m/s, the momentum is:
Momentum2 = 2.97 kg * (0.99 i + 5.94 j) m/s
Adding these two momenta together, we get the total momentum of the system:
Total momentum = Momentum1 + Momentum2
Substituting the values, we have:
Total momentum = 2.09 kg * (2.03 i - 2.92 j) m/s + 2.97 kg * (0.99 i + 5.94 j) m/s
Simplifying the calculation, we get:
Total momentum = (6.6247 i + 1.741 j) kg*m/s
Therefore, the total momentum of the system is (6.62 i + 1.74 j) kg*m/s.