An example of an elastic collision is The gravitational slingshot effect . A spacefship of mass 700 kg is moving at 11.0 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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This is probably what your big ball little ball problem was supposed to be about as well.

To estimate the final speed of the spacecraft after it is far enough to be considered free of Saturn's gravitational pull, we can apply the principles of conservation of momentum and conservation of kinetic energy.

1. Conservation of momentum:
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision.
In this case, the only external force acting on the system is the gravitational attraction of Saturn. So, during the swing around Saturn, no external forces are acting on the spacecraft, making this a perfect case for an elastic collision.

Therefore, we can write the equation for conservation of momentum as:
(m_spacecraft * v_initial_spacecraft) + (m_Saturn * v_initial_Saturn) = (m_spacecraft * v_final_spacecraft) + (m_Saturn * v_final_Saturn)

2. Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Since no external forces act during the swing around Saturn, the kinetic energy of the system is conserved.

Therefore, we can write the equation for conservation of kinetic energy as:
(0.5 * m_spacecraft * v_initial_spacecraft^2) + (0.5 * m_Saturn * v_initial_Saturn^2) = (0.5 * m_spacecraft * v_final_spacecraft^2) + (0.5 * m_Saturn * v_final_Saturn^2)

Now, let's substitute the given values into these equations to calculate the final speed of the spacecraft:

Given:
m_spacecraft = 700 kg
v_initial_spacecraft = 11.0 km/s (in the +x direction)
m_Saturn = 5.68 x 10^26 kg
v_initial_Saturn = -x km/s (the speed of Saturn is in the -x direction, but the value is not given)

Here, we don't know the value of the initial speed of Saturn, but we don't need it to find the final speed of the spacecraft.
We can solve the equations using symbolic variables to represent the unknown values.

Using a symbolic solver or a numerical method, we can rearrange the equations to solve for v_final_spacecraft and v_final_Saturn simultaneously. However, since the mass of Saturn is extremely large compared to the spacecraft, the final velocity of Saturn will be negligible. Therefore, we can assume v_final_Saturn is close to zero.

By solving the equations, we can find the final velocity of the spacecraft, v_final_spacecraft, which will be the answer to the question.