Two balls are approaching each other head-on. Their velocities are +11.9 and -10.5 m/s. Determine the velocity of the center of mass of the two balls if (a) they have the same mass and (b) if the mass of one ball (v = 11.9 m/s) is twice the mass of the other ball (v = -10.5 m/s).
To determine the velocity of the center of mass of the two balls, we need to apply the concept of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
(a) When the balls have the same mass:
Let's denote the mass of each ball as m.
The initial momentum of the system is given by:
Initial momentum = (mass of ball 1) * (velocity of ball 1) + (mass of ball 2) * (velocity of ball 2)
= m * 11.9 m/s + m * (-10.5 m/s)
= 1.9m m/s - 10.5m m/s
= -8.6m m/s
Since the balls are approaching head-on, their velocities have opposite directions. Therefore, we consider the negative sign for the second term.
Now, let's find the final momentum of the system. After the collision, both balls stick together and move with a common velocity.
Since the balls stick together, they have the same velocity after the collision. Let's denote this common velocity as V.
The final momentum of the system is given by:
Final momentum = (mass of ball 1 + mass of ball 2) * (velocity of the center of mass)
= (2m) * V
= 2mV
According to conservation of momentum, the initial momentum is equal to the final momentum:
Initial momentum = Final momentum
-8.6m = 2mV
Solving this equation for V, we get:
V = -4.3 m/s
Therefore, the velocity of the center of mass of the two balls, when they have the same mass, is -4.3 m/s.
(b) When the mass of one ball (v = 11.9 m/s) is twice the mass of the other ball (v = -10.5 m/s):
Let's denote the mass of the smaller ball as m and the mass of the larger ball as 2m.
Similarly, we can apply conservation of momentum to find the velocity of the center of mass.
The initial momentum of the system is given by:
Initial momentum = (mass of smaller ball) * (velocity of smaller ball) + (mass of larger ball) * (velocity of larger ball)
= m * (-10.5 m/s) + 2m * 11.9 m/s
= -10.5m m/s + 23.8m m/s
= 13.3m m/s
Now, let's find the final momentum of the system. After the collision, both balls stick together and move with a common velocity.
The final momentum of the system is given by:
Final momentum = (mass of smaller ball + mass of larger ball) * (velocity of the center of mass)
= (m + 2m) * (velocity of the center of mass)
= 3m * (velocity of the center of mass)
According to conservation of momentum, the initial momentum is equal to the final momentum:
Initial momentum = Final momentum
13.3m = 3m * (velocity of the center of mass)
Solving this equation for the velocity of the center of mass, we get:
Velocity of the center of mass = 13.3m / (3m) = 4.43 m/s
Therefore, the velocity of the center of mass of the two balls, when one ball has twice the mass of the other, is 4.43 m/s.