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Answer :
Using the conservation of momentum, the final velocity of the space probe after the engine fires is calculated to be approximately 100.2 m/s. The initial momentum of the system (space probe and exhaust) must equal the final momentum, allowing us to solve for the final velocity of the space probe. So the correct option is b) 100.2 m/s.
The problem presented involves the conservation of momentum in a system where a space probe fires its rocket engine to change its velocity. When the rocket ejects mass in the form of exhaust, momentum must be conserved because there are no external forces acting on the system. Therefore, the momentum of the probe and the exhaust must be equal and opposite before and after the engine fires, according to the conservation of momentum principle.
To find the final velocity of the space probe, we apply the conservation of momentum, m1 * v1 = m2 * v2 + [tex]m_{exhaust[/tex] * [tex]v_{exhaust[/tex], where m1 is the initial mass of the probe, v1 is the initial velocity, m2 is the final mass of the probe, v2 is the final velocity, [tex]m_{exhaust[/tex] is the mass of the exhaust, and [tex]v_{exhaust[/tex] is the exhaust velocity relative to the probe.
Here the initial mass of the probe (m1) is 6330 kg, the initial velocity (v1) is 99.0 m/s, the final mass (m2) is m1 - [tex]m_{exhaust[/tex], the mass of the exhaust ([tex]m_{exhaust[/tex]) is 65 kg, and the exhaust velocity ([tex]v_{exhaust[/tex]) is 229 m/s relative to the probe.
Using the conservation of momentum, 6330 kg * 99.0 m/s = (6330 kg - 65 kg) * v2 + 65 kg * (-229 m/s), where we use a negative for the exhaust velocity because it goes in the opposite direction of the probe's initial velocity.
Solving for v2, we obtain a final velocity of approximately 100.2 m/s for the space probe. Hence, the correct option is b) 100.2 m/s.
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