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Say you want to fly to Pluto from Earth and orbit it.

So you're going to launch it from Earth with such a speed that as it approaches Pluto, it's traveling (roughly) the same linear speed of Pluto.

How long should it take to get there?

So I was thinking of using the classical $s=(at^2)/2$ and solve for $t$.

But $a$ is variable (as you're flying from the Sun, the Force changes by a few orders of magnitude)

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To travel directly to Pluto you'd use a Hohmann transfer orbit something like this:

Hohmann transfer orbit

(not to scale - Pluto is about 40 times farther from the Sun than Earth is!)

Starting in orbit around the Earth the spaceship fires its motors to increase its speed and put it into an elliptical orbit with the aphelion at Pluto. Then when the spaceship reaches Pluto it would fire its engines again to match speed with Pluto.

To calculate the required orbit exactly would be a lot of work, but we can use an approximation. The orbital period of an elliptical orbit is given by:

$$ T = 2\pi\sqrt{\frac{a^3}{GM}} $$

where $M$ is the mass of the Sun and $a$ is the semi-major axis. Pluto orbits at around 40 AU, so the Earth-Pluto distance as drawn is 41 AU and the semi-major axis is $a \approx$ 20.5 AU.

That gives us $T \approx 2.9 \times 10^9$ seconds or about 93 years. The time to reach Pluto would be half this or about 46.5 years.

Note that a Hohmann transfer orbit only fires the rocket motor twice: once to leave Earth's orbit and once to match speed and go into orbit with the destination. Given that spaceships can't carry unlimited amounts of fuel this is the normal procedure. If you're not fussed about going into orbit round Pluto you could use some sort of multiple stage motor to leave Earth orbit on a hyperbolic trajectory, and this would reduce the travel time. The problem is that you'd whizz past Pluto - just as New Horizons did in fact. The travel time would then just depend on how big a multistage propulsion system you managed to build.

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