A spaceship of mass 2.3×10^6 kg is cruising at a speed of 6.0×10^6 m/s when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of 5.2×10^5 kg, is blown straight backward with a speed of 2.0×10^6 m/s. A second piece, with mass 8.4×10^5 kg, continues forward at 9.0×10^5 m/s. What is the speed of the third piece?
p=mv=2.3•10⁶•6•10⁶=13.8•10¹² kg•m/s.
p₁= m₁v₁= 5.2•10⁵•2•10⁶=10.4•10¹¹kg•m/s.
p₂=m₂v₂=8.4•10⁵•9•10 ⁵=75.6•10¹ºkg•m/s.
p₃=m₃v₃.
m₃=m-m₁-m₂=2.3•10⁶ -5.2•10⁵-8.4•10⁵=(23-5.2-8.4)•10⁵=9.4•10⁵ kg
Law of conservation of linear momentum
p=- p₁+p₂+p₃
p₃=p- p₁+p₂.
v₃=(p+ p₁-p₂)/m₃=
=( 13.8•10¹²+10.4•10¹¹-75.6•10¹º)/ 9.4•10⁵=
=(1380+104-75.6) 10¹º/9.4•10⁵=149.8•10⁵=
=1.498•10⁷ m/s
To find the speed of the third piece, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.
The momentum of an object is given by the product of its mass and velocity. Therefore, the momentum of the spaceship before the explosion can be calculated by multiplying its mass by its velocity:
Momentum of spaceship before explosion = (mass of spaceship) × (velocity of spaceship)
Momentum of spaceship before explosion = (2.3×10^6 kg) × (6.0×10^6 m/s)
Next, we will calculate the total momentum after the explosion. Since the spaceship breaks into three pieces, we need to calculate the momentum of each piece independently and then add them up.
First, let's calculate the momentum of the backward piece:
Momentum of backward piece = (mass of backward piece) × (velocity of backward piece)
Momentum of backward piece = (5.2×10^5 kg) × (2.0×10^6 m/s)
Similarly, let's calculate the momentum of the forward piece:
Momentum of forward piece = (mass of forward piece) × (velocity of forward piece)
Momentum of forward piece = (8.4×10^5 kg) × (9.0×10^5 m/s)
Now, to find the momentum of the third piece, we can subtract the sum of the backward and forward momenta from the total momentum before the explosion:
Momentum of third piece = Momentum of spaceship before explosion - (Momentum of backward piece + Momentum of forward piece)
Finally, to find the speed of the third piece, we divide its momentum by its mass:
Speed of third piece = Momentum of third piece / Mass of third piece
Now you have all the information you need and can plug in the given numerical values to find the final answer.