A spacecraft approaches a gas giant planet at a speed of 10,000 m/s and uses a gravitational slingshot maneuver to increase its velocity. Calculate the spacecraft's speed after the slingshot, considering the planet's mass.
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To calculate the spacecraft's speed after the slingshot maneuver, we need to consider the conservation of momentum and energy.
The spacecraft's initial momentum before the slingshot is given by:
p1 = m * v1
where m is the spacecraft's mass and v1 is its initial velocity (10,000 m/s).
During the slingshot maneuver, the spacecraft gains momentum from the gravitational interaction with the planet. Momentum is conserved in this interaction, resulting in:
p1 + p2 = p1' + p2'
where p2 is the planet's momentum before the interaction and p1' and p2' are the spacecraft's and planet's momentum after the interaction.
Since the planet's mass is significantly larger than the spacecraft's mass, we can assume that its velocity before and after the slingshot remains negligible (p2 ≈ 0), so the equation simplifies to:
p1 = p1' + p2'
The gravitational slingshot maneuver can be treated as an elastic collision, conserving both momentum and kinetic energy. Therefore, the kinetic energy of the spacecraft after the slingshot is the same as its initial kinetic energy.
Using the kinetic energy equation:
KE = (1/2) * m * v^2
where KE is the kinetic energy and v is the velocity, we can write:
(1/2) * m * v1^2 = (1/2) * m * v2^2
Canceling the mass from both sides of the equation gives:
v1^2 = v2^2
Taking the square root of both sides of the equation:
v1 = v2
Therefore, the spacecraft's velocity after the slingshot (v2) is still 10,000 m/s.
In conclusion, the spacecraft's speed remains unchanged after the slingshot maneuver and stays at 10,000 m/s.
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