Can you solve with relative motion?

Sending spaceships to the outer regions of the solar system requires large energies. The slingshat effect is a frequently used method for speeding up spaceships without using much rocket fuel. To use this elfect the spaceship is diverted to pass around one of the inner planets $($such as Venus$).$ If the spacoship is sent in the right direction, it will reach to greater speeds after it passes around the planet. You will see the effect at play in this question.

A spaceship is sentwith an initial velocity of vec$(v{)}_S=(20*hat(j)+15*hat(i))k\frac{m}{s}$ cowards a planet The planet is moving with velocity vec$(v)p=30$hat$\frac{km}{s}.$ After the encounter with the planet, the spaceship emerges to he moving along the I $x$ direction $($with a final velocity. of the form vec$(v{)}_S=v_S^{\text{'}}$hat$(i).$ Find the final speed of the spaceship in units of $k\frac{m}{s}.$

$v_S^{\text{'}}=k\frac{m}{s}.$

Hint $1$: This is an elastic "collision". When you hear the ward collision, you usually imagine something that happens quickiy: in contrast this collision happens over days and perhops even weeks. But, you can use the same equations to treat all.

Hint $2$: This is really unnecessary to state, but sometimes lam abliged to state the obvious$)$: Assume that the mass of the planet is much larger than the mass of the spaceship.