Final answer:
The fourth piece of the exploded rock has a mass of 120 kg and will move to the right at a velocity of 21.167 m/s to conserve the total momentum.
Step-by-step explanation:
The problem involves the conservation of momentum, where the total momentum before the explosion must equal the total momentum after the explosion. As the rock has exploded into four pieces, we know that the momentum of all pieces combined must add up to zero, as the rock was presumably at rest before the explosion.
The first piece is 500-kg moving to the left at 2 m/s, the second piece is 300 kg moving to the right at 1 m/s, and the third piece is 80 kg moving to the left at 23 m/s. To find the Fourth piece's mass, we subtract the total mass of the first three from 1000-kg: 1000 kg - (500 kg + 300 kg + 80 kg) = 120 kg. To find its velocity, we set up the momentum equation:
Momentum of first piece:
(500 kg)(-2 m/s) = -1000 kg·m/s
Momentum of second piece:
(300 kg)(1 m/s) = 300 kg·m/s
Momentum of third piece:
(80 kg)(-23 m/s) = -1840 kg·m/s
The total momentum of these three pieces is -1000 kg·m/s + 300 kg·m/s - 1840 kg·m/s = -2540 kg·m/s. To maintain zero total momentum, the fourth piece must have an opposite momentum of +2540 kg·m/s. The velocity (v) of the fourth piece is given by the momentum divided by its mass (120 kg): v = 2540 kg·m/s / 120 kg = 21.167 m/s to the right (as it must balance the direction of the total momentum of the other pieces).
Therefore, the fourth piece has a mass of 120 kg and will move to the right at 21.167 m/s.