To determine the velocity of the center of mass of the two balls, we can use the concept of the conservation of momentum. The principle of the conservation of momentum states that the total momentum of a system remains constant if no external forces act on it.
Let's start with part (a), where the masses of both balls are equal.
(a) The velocity of the center of mass (Vcm) can be calculated using the equation:
Vcm = (m1 * v1 + m2 * v2) / (m1 + m2)
Where:
- Vcm is the velocity of the center of mass.
- m1 and m2 are the masses of the balls.
- v1 and v2 are the velocities of the balls.
Since the masses of both balls are the same, let's consider them as m1 = m2 = m. Also, the given velocities are +11.9 m/s and -10.5 m/s.
Vcm = (m * 11.9 + m * (-10.5)) / (m + m)
Vcm = (11.9 - 10.5) / 2
Vcm = 1.4 / 2
Vcm = 0.7 m/s
Therefore, when the two balls have the same mass, the velocity of the center of mass is 0.7 m/s.
Now let's move on to part (b), where the mass of one ball is twice the mass of the other ball.
In this case, let's consider the mass of one ball as m1 and the mass of the other ball as m2, where m2 = 2m1.
To find the velocity of the center of mass, we can use the same equation:
Vcm = (m1 * v1 + m2 * v2) / (m1 + m2)
Substituting the given values, m1 = m and m2 = 2m, and v1 = 11.9 m/s, and v2 = -10.5 m/s, we have:
Vcm = (m * 11.9 + 2m * (-10.5)) / (m + 2m)
Vcm = (11.9m - 21m) / 3m
Vcm = -9.1m / 3m
Vcm = -3.03 m/s
Therefore, when the mass of one ball is twice the mass of the other ball, the velocity of the center of mass is -3.03 m/s.