A spaceship of mass 2.0×106kg is cruising at a speed of 4.6×106m/s when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of 5.2×105kg , is blown straight backward with a speed of 2.2×106m/s . A second piece, with mass 7.7×105kg , continues forward at 1.3×106m/s .
1)What is the speed of the third piece? Assume that the initial speed of the ship is positive.
To find the speed of the third piece, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion should be equal to the total momentum after the explosion.
Before the explosion, the momentum of the spaceship is given by:
Momentum = mass × velocity
The momentum of the spaceship before the explosion is given by:
Momentum_before_explosion = (mass of the spaceship) × (velocity of the spaceship)
Momentum_before_explosion = (2.0×10^6 kg) × (4.6×10^6 m/s)
After the explosion, the momentum of the first piece (blown straight backward) is given by:
Momentum_first_piece = (mass of the first piece) × (velocity of the first piece)
Momentum_first_piece = (5.2×10^5 kg) × (2.2×10^6 m/s)
The momentum of the second piece (continuing forward) is given by:
Momentum_second_piece = (mass of the second piece) × (velocity of the second piece)
Momentum_second_piece = (7.7×10^5 kg) × (1.3×10^6 m/s)
According to the principle of conservation of momentum, the total momentum before the explosion should be equal to the total momentum after the explosion:
Momentum_before_explosion = Momentum_first_piece + Momentum_second_piece + Momentum_third_piece
To find the momentum of the third piece, we rearrange the equation:
Momentum_third_piece = Momentum_before_explosion - (Momentum_first_piece + Momentum_second_piece)
Substituting the given values, we have:
Momentum_third_piece = (2.0×10^6 kg) × (4.6×10^6 m/s) - ((5.2×10^5 kg) × (2.2×10^6 m/s) + (7.7×10^5 kg) × (1.3×10^6 m/s))
Simplifying the equation gives:
Momentum_third_piece ≈ 9.2 × 10^12 Ns
To find the speed of the third piece, we divide the momentum by its mass:
Speed_third_piece = Momentum_third_piece / (mass of the third piece)
Speed_third_piece = (9.2 × 10^12 Ns) / (mass of the third piece)
Speed_third_piece ≈ (9.2 × 10^12 Ns) / (mass of the third piece)
Substituting the given mass of the third piece:
Speed_third_piece ≈ (9.2 × 10^12 Ns) / (mass of the third piece = 7.7 × 10^5 kg)
Calculating this gives:
Speed_third_piece ≈ 1.19 × 10^7 m/s
Therefore, the speed of the third piece is approximately 1.19 × 10^7 m/s.