The common speed of the combined packages after the collision is 0 m/s. The package of mass m rebounds to a height of 5.0 m.
When the package of mass m slides down the frictionless chute and collides with the package of mass 2m, they stick together and move as one. To find their common speed after the collision, we can apply the principle of conservation of momentum. Momentum is conserved in an inelastic collision, so the total momentum before the collision equals the total momentum after the collision.
Before the collision, the package of mass m is released from rest, so its initial velocity is 0 m/s. The package of mass 2m is at the bottom of the chute and is not moving, so its initial velocity is also 0 m/s.
After the collision, the two packages stick together. Let's denote their common velocity as v. The total mass of the system is m + 2m = 3m. Therefore, the total momentum after the collision is (3m)(v) = 3mv.
Since momentum is conserved, we can set the initial momentum equal to the final momentum:
0 + 0 = 3mv.
Solving for v, we get v = 0 m/s.
Therefore, the common speed after the collision is 0 m/s.
To find the height to which the package of mass m rebounds, we can use the principle of conservation of mechanical energy. In this case, we assume the collision is perfectly elastic.
Before the collision, the package of mass m is at a height of 5.0 m. The potential energy at this height is given by mgh, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
The total mechanical energy before the collision is equal to the sum of the potential energy and the kinetic energy of the package of mass m. Since the package is at rest, the initial kinetic energy is 0 J.
After the collision, the package of mass m rebounds to a height h. At this height, its potential energy is given by mgh.
The total mechanical energy after the collision is equal to the sum of the potential energy and the kinetic energy of the package of mass m. Since the final velocity is 0 m/s at this height (reached momentarily at the highest point during the rebound), the final kinetic energy is 0 J.
Since mechanical energy is conserved, we can set the initial mechanical energy equal to the final mechanical energy:
mgh + 0 = mgh + 0.
Simplifying the equation, we find h = 5.0 m.
Therefore, the package of mass m rebounds to a height of 5.0 m.