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 the new velocity of the stolen Klingon Bird-of-Prey (v_new) in terms of u and v, we need to apply the principle of conservation of momentum.
The total momentum of the system before traveling around the sun is given by the sum of the momentum of the spaceship (m * v) and the momentum of the sun (M * u), where m is the mass of the spaceship and M is the mass of the sun.
After traveling around the sun, the spaceship changes its direction of motion and gains a new velocity, v_new. The momentum of the system after the maneuver is given by the sum of the momentum of the spaceship (m * v_new) and the momentum of the sun (-M * u) since their velocities are now in opposite directions.
According to the principle of conservation of momentum, the total momentum before and after the maneuver should be the same. Therefore, we can equate the total momentum before and after:
m * v + M * u = m * v_new - M * u
Rearranging the equation, we can solve for v_new:
m * v_new = m * v + M * u + M * u
v_new = (m * v + 2M * u) / m
Since the mass of the sun (M) is much larger than the mass of the spaceship (m), we can neglect the term mM compared to mv:
v_new ≈ (m * v + 2M * u) / m
Finally, the expression for v_new in terms of u and v is:
v_new ≈ v + (2M * u) / m