Determine the final velocity of a rocket that expels mass at a constant rate of 1 kg/s, starting from rest and experiencing no external forces other than the rocket's propulsion. The initial mass of the rocket is 10,000 kg.
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To find the final velocity of the rocket, we can use the principle of conservation of momentum. Since there are no external forces acting on the rocket other than its propulsion, the change in momentum can be determined solely by the change in mass.
Let's denote the initial mass of the rocket as M1 and the final mass as M2. The change in mass will be ΔM = M2 - M1 = -1 kg/s * t, where t is the time in seconds.
According to the principle of conservation of momentum, the change in momentum is equal to the mass times the change in velocity: Δp = ΔM * v, where v is the change in velocity.
Since the rocket starts from rest, its initial momentum is zero: p1 = 0 kg m/s.
The final momentum is given by: p2 = ΔM * v.
Using the conservation of momentum, we can equate the initial and final momentum:
0 = ΔM * v.
Simplifying the equation gives:
0 = -1 kg/s * t * v.
Since the rocket expels mass at a constant rate, the time t and the change in mass ΔM are related by ΔM = -t.
Therefore, the equation becomes:
0 = (-1 kg/s * t) * v.
To find the final velocity, we solve for v:
v = 0 / (-1 kg/s * t) = 0.
The final velocity of the rocket is 0 m/s.
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