In 2286, Admiral Kirk and his crew were forced to use the sling
shot effect in a stolen Klingon Bird-of-Prey to travel back in time
to the late 20th century to retrieve two humpback whales. The
stolen Klingon Bird-of-Prey traveled towards the sun at a velocity
of v, while the sun was moving towards them at a velocity of u,
then traveled around the sun (using the sun's gravitational field)
such that the stolen Klingon Bird-of-Prey was now moving in the
opposite direction from whence it started, with a new velocity,
v
new
. Find v
new
in terms of u and v, assuming the mass of the sun is
much larger than the mass of the spaceship.
To find v_new in terms of u and v, we can use the principle of conservation of momentum. Conservation of momentum states that the total momentum of an isolated system remains constant before and after an event.
In this scenario, we can consider the stolen Klingon Bird-of-Prey and the sun as an isolated system. Before traveling around the sun, the total momentum of the system is the sum of the momentum of the spaceship and the momentum of the sun.
The momentum of an object is given by its mass multiplied by its velocity (p = mv). Since the mass of the sun is much larger than the mass of the spaceship, we can neglect the contribution of the sun's velocity to the equation.
Before traveling around the sun, the momentum of the system is:
P_initial = (mass of the spaceship * velocity) + (mass of the sun * velocity)
After traveling around the sun, the stolen Klingon Bird-of-Prey is moving in the opposite direction with a new velocity, v_new. The momentum of the system is now:
P_final = (mass of the spaceship * v_new) + (mass of the sun * (-u))
Since the total momentum of an isolated system remains constant, P_initial = P_final.
Therefore,
(mass of the spaceship * velocity) + (mass of the sun * velocity) = (mass of the spaceship * v_new) + (mass of the sun * (-u))
Now we can rearrange the equation to solve for v_new:
mass of the spaceship * v_new = (mass of the spaceship * velocity) + (mass of the sun * velocity) + (mass of the sun * u)
Divide both sides by the mass of the spaceship to isolate v_new:
v_new = (mass of the spaceship * velocity + mass of the sun * velocity + mass of the sun * u) / mass of the spaceship
Notice that the mass of the spaceship cancels out from both sides of the equation, leaving:
v_new = velocity + (mass of the sun * velocity + mass of the sun * u)/mass of the spaceship
Since the mass of the sun is much larger than the mass of the spaceship, we can approximate mass of the sun / mass of the spaceship as infinity. As a result, we can neglect the contribution of (mass of the sun * velocity + mass of the sun * u) to the equation.
Finally, we have:
v_new ≈ velocity
Therefore, the new velocity after traveling around the sun, v_new, is approximately equal to the initial velocity, v.