In 2286, Admiral Kirk and his crew were forced to use the slingshot 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 vs 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 stolen Klingon Bird-of-Prey was now moving in the opposite direction from whence it started, with a new velocity, vnew. Find vnew in terms of u and v,assuming the mass of the sun is much larger than the mass of the spaceship

Question

In 2286, Admiral Kirk and his crew were forced to use the slingshot 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 vs 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 stolen Klingon Bird-of-Prey was now moving in the opposite direction from whence it started, with a new velocity, vnew. Find vnew in terms of u and v,assuming the mass of the sun is much larger than the mass of the spaceship

Answer 1

To find the new velocity, vnew, of the stolen Klingon Bird-of-Prey after traveling around the sun, we can use the conservation of momentum.

Let's consider the initial momentum of the spaceship, which is given by the product of its mass (m) and its initial velocity (vs). Since the spaceship's mass remains constant, we can only consider the change in velocity.

Initially, the spaceship is moving towards the sun at a velocity of vs. The sun is also moving towards the spaceship at a velocity of u. Therefore, the relative velocity between the spaceship and the sun is the difference between their velocities:

Relative velocity = vs - u

Now, when the spaceship travels around the sun using the sun's gravitational field, its final velocity, vnew, is in the opposite direction from its initial velocity, vs. So, we have:

vnew = -Relative velocity

Substituting the relative velocity, we get:

vnew = -(vs - u)
= -vs + u

Therefore, the new velocity of the stolen Klingon Bird-of-Prey after traveling around the sun is given by vnew = -vs + u, where u is the velocity of the sun towards the spaceship and vs is the initial velocity of the spaceship towards the sun.

Note: This answer assumes that the only significant force acting on the spaceship is the gravitational force from the sun, and the mass of the sun is much larger than the mass of the spaceship.