An example of an elastic collision is The gravitational slingshot effect . A spacefship of mass 850 kg is moving at 9.5 km/s in the +x direction. It approaches the planet Saturn, mass 5.68 x10 26kg , which is moving in the –x- direction as shown in the figure. The gravitational attraction of Saturn accelerates the spaceship as it approaches and causes to swing around the planet and heads off in the same direction as Saturn. Several probes such as Voyager have used this to save on “gas”. Estimate the final speed of the spacecraft after it is far enough to be considered free of Saturn’s gravitational pull. Assume that the speed of Saturn is not affect due to its large mass. Express your answer in km/s. HINT: perfectly ELASTIC COLLISION.

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u + 2 U

To estimate the final speed of the spacecraft after it is far enough to be considered free of Saturn's gravitational pull, we can analyze this situation as an elastic collision.

In an elastic collision, both momentum and kinetic energy are conserved before and after the collision. Since the mass of Saturn is much larger than the mass of the spacecraft, we can neglect any change in Saturn's velocity.

Let's break down the problem into two phases: the approach phase and the departure phase.

Approach Phase:
- The mass of the spaceship, m₁ = 850 kg
- Initial velocity of the spaceship, u₁ = 9.5 km/s (in the +x direction)
- The mass of Saturn, m₂ = 5.68 x 10^26 kg (which we assume to have no change in velocity)

In the approach phase, the gravitational attraction of Saturn accelerates the spaceship as it approaches. The velocity of the spaceship changes in direction but still moves in the x-direction. We need to find the velocity of the spaceship, v₁, just before the collision (when it swings around Saturn).

To calculate v₁, we'll use the principle of conservation of momentum in the x-direction:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Since Saturn's velocity, u₂, is in the -x direction, it is -9.5 km/s.

Substituting the values into the equation, we have:

(850 kg)(9.5 km/s) + (5.68 x 10^26 kg)(-9.5 km/s) = (850 kg)(v₁) + (5.68 x 10^26 kg)(v₂)

Solving for v₁, we get:

v₁ = [(850 kg)(9.5 km/s) + (5.68 x 10^26 kg)(-9.5 km/s)] / (850 kg)

Departure Phase:
After the swing around Saturn, the spaceship heads off in the same direction as Saturn. We assume this phase is a perfectly elastic collision.

Since momentum and kinetic energy are conserved, we can use the equation for conservation of momentum to find the final velocity, v₂, of the spaceship after the collision.

m₁v₁ + m₂v₂ = m₁u₁ + m₂u₂

Since the spaceship and Saturn move in the same direction after the collision, v₂ is positive.

Substituting the values into the equation, we get:

(850 kg)(v₁) + (5.68 x 10^26 kg)(v₂) = (850 kg)(9.5 km/s) + (5.68 x 10^26 kg)(-9.5 km/s)

Solving for v₂, we can find the final velocity of the spaceship.

Since there is no external force acting on the spaceship after it is considered free from Saturn's gravitational pull, the final velocity of the spaceship would be its final speed.

Using these equations and plugging in the values, you can calculate the final speed of the spacecraft.