A spaceship of mass 2.1×10^6 kg is cruising at a speed of 5.0×10^6 m/s when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of 4.8×10^5 kg , is blown straight backward with a speed of 2.3×10^6 m/s . A second piece, with mass 8.4×10^5 kg , continues forward at 1.0×10^6 m/s .
What is the speed of the third piece? Assume that the initial speed of the ship is positive
momentum is conserved, and since two pieces are in the direction of initial movement, the third cannot be in another track, it has to be in the same direction
Momentum:
2.1E6*5E6=4.8E5*(-2.3E6)+8.4E5*1E6+(2.1E6-4.8E5-8.4E5)V
solve for V
To find the speed of the third piece, we can use the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event.
The momentum (p) of an object is given by the product of its mass (m) and velocity (v):
p = mv
Since the initial speed of the ship is positive, the total momentum before the breakup is the sum of the momentum of each section moving in the positive direction.
Let's assign variables:
m1 = mass of first section = 4.8×10^5 kg
v1 = velocity of first section = -2.3×10^6 m/s (negative since it is moving backward)
m2 = mass of second section = 8.4×10^5 kg
v2 = velocity of second section = 1.0×10^6 m/s (positive as it continues forward)
m3 = mass of third section = ?
v3 = velocity of third section = ?
According to the principle of conservation of momentum:
Total momentum before = Total momentum after
(mass of ship) * (initial speed of ship) = (m1 * v1) + (m2 * v2) + (m3 * v3)
Now we can plug in the given values:
(2.1×10^6 kg) * (5.0×10^6 m/s) = (4.8×10^5 kg * -2.3×10^6 m/s) + (8.4×10^5 kg * 1.0×10^6 m/s) + (m3 * v3)
Simplifying the equation:
1.05×10^13 kg m/s = -1.104×10^12 kg m/s + 8.4×10^11 kg m/s + m3 * v3
Rearranging the equation to solve for m3 * v3:
m3 * v3 = 1.05×10^13 kg m/s - (-1.104×10^12 kg m/s + 8.4×10^11 kg m/s)
m3 * v3 = 1.05×10^13 kg m/s + 2.64×10^12 kg m/s
m3 * v3 = 1.316×10^13 kg m/s
Now we can solve for v3 by dividing both sides of the equation by m3:
v3 = (1.316×10^13 kg m/s) / m3
Therefore, the speed of the third piece depends on the mass of the third piece, which is not given in the question.
To find the speed of the third piece, we can use the conservation of momentum. The total momentum before the explosion must equal the total momentum after the explosion.
The total momentum before the explosion can be calculated by multiplying the mass of the entire spaceship by its velocity:
Momentum before = mass of spaceship × velocity of spaceship
Momentum before = (2.1×10^6 kg) × (5.0×10^6 m/s)
Momentum before = 1.05×10^13 kg⋅m/s
The total momentum after the explosion will be the sum of the momenta of each individual piece. We already know the mass and velocity of two pieces, so let's calculate their momenta first.
Momentum of first piece = mass of first piece × velocity of first piece
Momentum of first piece = (4.8×10^5 kg) × (2.3×10^6 m/s)
Momentum of first piece = 1.104×10^12 kg⋅m/s
Momentum of second piece = mass of second piece × velocity of second piece
Momentum of second piece = (8.4×10^5 kg) × (1.0×10^6 m/s)
Momentum of second piece = 8.4×10^11 kg⋅m/s
Now, we can find the momentum of the third piece by subtracting the total momentum of the first two pieces from the total initial momentum.
Momentum of third piece = Momentum before - Momentum of first piece - Momentum of second piece
Momentum of third piece = 1.05×10^13 kg⋅m/s - 1.104×10^12 kg⋅m/s - 8.4×10^11 kg⋅m/s
Momentum of third piece = 8.46×10^12 kg⋅m/s
Finally, we can find the speed of the third piece by dividing its momentum by its mass:
Speed of third piece = Momentum of third piece / Mass of third piece
Speed of third piece = (8.46×10^12 kg⋅m/s) / (mass of third piece)
Substituting the mass of the third piece:
Speed of third piece = (8.46×10^12 kg⋅m/s) / (mass of third piece) = (8.46×10^12 kg⋅m/s) / (2.1×10^6 kg) = 4.0286×10^6 m/s
Therefore, the speed of the third piece is approximately 4.03×10^6 m/s.