Why do many comets & asteroids keep moving through the solar system (for centuries), after they were dislodged from their parent bodies after a cosmic event/explosion?
But a space-shuttle traveling will need constant supply of fuel to do so and maintain trajectory and velocity.
Assuredly assuming that comets & asteroids don’t have pilots to help them slingshot around other planets.
Your assumptions are incorrect.
A space ship (once it has escaped Earth) will also continue travelling - in fact it will only need fuel to change its trajectory and velocity outwith gravitational effects from other bodies.
You will need sufficient fuel or power for life support, if you have a crewed ship, and a reserve of fuel to manoeuvre at your destination, but in the middle there is just a lot of coasting, with minor attitude adjustments . See, for example, Perseverance's course info:
The cruise phase begins after the spacecraft separates from the rocket, soon after launch. The spacecraft departs Earth at a speed of about 24,600 mph (about 39,600 kph). The trip to Mars will take about seven months and about 300 million miles (480 million kilometers). During that journey, engineers have several opportunities to adjust the spacecraft’s flight path, to make sure its speed and direction are best for arrival at Jezero Crater on Mars. The first tweak to the spacecraft’s flight path happens about 15 days after launch.
Spaceships will keep travelling in their orbits just the same as comets or asteroids, without fuel. The exception is that spacecraft in low orbit are affected by the upper fringe of the atmosphere, and need a slight boost every few months.
Spaceships need fuel to change their course or to fly from the ground into space and back. Typically they only run their engines for a few minutes, and then coast for months or years. Comets and asteroids and planets don't really change direction very much except when they pass by something with enough gravity to redirect them (which spacecraft often do as well to save fuel).
If there were no forces on an object, then Newton's First Law tells us that a stationary object would remain stationary, and a moving object will keep moving. So a moving comet or asteroid keeps moving.
When there is a force on an object, Newton's Second Law essentially tells us that the object will speed up, slow down, and/or change directions. Rockets take advantage of this to speed up at the start of their mission, change directions for course corrections, and slow down at their destination. They need fuel to do these three things; however, most of the time they are simply coasting just like comets and asteroids.
The gravity of the Sun, planets, and moons are other forces that can act on objects in space. However, the force of gravity is usually rather weak in space, so its effects happen gradually. The object will slowly speed up, slow down, and/or change directions, creating a curved path. But gravity making an object in space come to a full stop isn't something that gravity does.
So comets, asteroids, and spacecraft that are in motion will remain in motion, although along the curved path created by gravity. Rocket fuel is needed to set the spacecraft into motion, to make course corrections, and to slow down at the destination, but fuel is not needed during most of the trip, when the spacecraft coasts.
Back in the 1600s, Isaac Newton discovered maths for the movement of objects which matched observations, especially the movement of objects in space where there is no friction. We call them Newton's Laws of Motion.
Newton's First Law is that you only need to apply a force to change your speed, not to continue at the same speed.
The problem is how accurately you can start your trajectory. The accuracy needed to get there is literally unmeasurable. As a result, some course corrections in the way are needed.. They're usually small, but they're always going to happen.
The key part of that question is "speed up". You could just put in a single big hit of acceleration and then let your spaceship basically coast all the way to the destination, just tweaking the trajectory slightly. That's what we've done so far.
We are discovering and developing a range of thruster technologies though which can apply very small amounts of force for a long time without needing much fuel. They're already used for altitude adjustments on satellites, where the tiny amount of friction from the outer atmosphere slowly degrades their orbits. The idea for longer voyages is that applying a small force continuously for a long time can eventually get you up to higher speeds than you could reach with a single powerful burst of acceleration. A Ferrari accelerating hard for 4s would get up to 60mph. Your grandmother's little hatchback accelerates much more slowly, but it can comfortably get up to 100mph if you give it long enough. That'll get you to your destination faster. . Of course there's no brakes in space, so the second half of the journey is the engine pushing the same amount in the opposite direction to slow you down.
They're using the same principle of applying a continuous force to get them there faster. You'll notice the spaceships flipping end-to-end, and that their exhaust plume is often in the opposite direction to where they're going, because they're on the deceleration part of getting there.
More interestingly, their (fictional) drive is also solving a standard problem of space travel - how you deal with not having gravity. Our bodies don't deal with that well, and the ISS has a range of complicated exercise systems for that. A common proposal is to spin the spacecraft to give a centripetal force equivalent to gravity. You'll see this on all those SF films with space stations with rings. (And what they mostly get wrong, by the way, is that the outer surface of the ring is the floor. You often see windows in them, and people looking out. Nope.) The problem here is that making course corrections from a spinning ship is mathematically hard.
There's another option though. Gravity is just a force. If you run your thruster continuously to provide 1G acceleration, then it'll feel just like home, and the back of the spaceship is "down". We don't yet have a thruster which can do this and keep an acceptable fuel consumption, but the science part of this is sound. It's the existence of this drive which makes it science fiction. Ironically, what The Expanse gets wrong isn't the fact of having gravity, it's having things floating as if there wasn't gravity!
Space is mostly empty, so there is nothing stopping or slowing a space ship until it encounters an orbital body. It’s pretty much like throwing a ball in vacuum. There is just no reason for it to stop.
At the right trajectory a space ship could travel (“coast”) forever.
In space you only need fuel to change your velocity. Which is also the reason why “mileage” for space ships is stated as “delta-v” i.e. the total change in velocity one can achieve with the remaining fuel reserves. Space ships only need fuel for the initial launch, course corrections and to slow at the destination.
A greater understanding of the physics involved is required here, both in the initial question and in many of the above answers, which I don't feel have adequately addressed the issue. Objects in space will orbit around the largest nearby gravitational source (although forces from other distant objects still play a factor). Planets and comets orbit the sun in heliocentric orbits, while most satellites orbiting the earth (including the moon and some space dust) are in geocentric orbits.
Most orbits are elliptical to some degree: orbits such as the earth around the sun are mostly circular, while comets are typically highly elliptical.
It is more convenient to use a polar coordinate system when examining orbits. More detail can be found Here, however the basics are as follows:
Focusing on a specific object as an example (such as the international space station), two forces are responsible for the curved nature of the orbit. Namely:
A normal force. This is the acceleration due to gravity, and is treated as a straight line between the satellite and the earth. It is this force which is responsible for the orbit being curved around the body at its center.
A tangential force. This is in the direction of travel.
In the case of a satellite launched from earth, the rocket initially accelerates vertically from the surface. However shortly after launch, the rocket curves over, until (once in space) it is flying parallel to the earth's surface. It then continues to accelerate until the tangential speed is high enough to prevent it from falling back to earth. Typically once in orbit, the engines are shut down: orbiting is purely due to conservation of momentum: I'll get to why this isn't entirely true below, but this is sufficient for now.
If the rocket keeps accelerating (due to engine thrust) the orbit around the earth will become more elliptical, until eventually it reaches escape velocity and is travelling fast enough to escape earth's orbit and become heliocentric. Once orbiting the sun, the engines are again shut down: orbit is once again purely due to conservation of momentum.
Such a satellite would be orbiting the sun, but with an orbit which would be very similar to the earth's. If the satellite/probe then wants to travel to another planet (such as mars) it will again accelerate (engines) to enlarge the orbit until it's close enough to mars to be influenced by that planet's gravity. To actually orbit, or land on mars, it then needs to decelerate (usually by turning until the engines face forward or by using retro-rockets) until it enters a mars orbit. Perhaps a better explanation (with pictures) is here.
At this point, hopefully this explains the larger celestial bodies: both planets and comets orbit the sun, although the latter usually have highly elliptical orbits. The 'curved' nature of both orbits is entirely due to the sun's gravity; neither planets nor comets need any thrust to continue on their path.
Similarly with rockets/satellites/the space shuttle, no additional engine thrust is needed to continue orbiting the earth: any movies showing spaceships with engine glow/flames are purely there for special effects (unless the craft is actually adjusting its orbit).
As I mentioned above, this explanation neglects one factor: drag. Around most planets (particularly the earth), there is an atmosphere, consisting of gas particles. In the case of earth, 'space' is typically defined as about 100km above the surface (the Kármán line), however the transition from 'atmosphere' to 'space' isn't a hard line; atmospheric particles around the earth exist as high as the edge of the thermosphere, about 690km above the surface. This is actually above the orbit of the moon, and well above the height of most geocentrically orbiting satellites (such as the ISS).
The significance of this is that as the satellite orbits, it collides with these atmospheric particles. These collisions produce drag, slowly reducing the tangential velocity, and causing the orbit to slowly decay until it re-enters the atmosphere. This is the reason that satellites eventually fall back to earth: the first artificial, man-made satellite (Sputnik 1) only stayed in space for about 3 months before it was burned up re-entering the atmosphere.
So this is where the caveat is: all satellites around the earth (below about 690km) experience drag and have decaying orbits. In the case of the ISS (orbiting approximately 400km away), the orbit decays, and so the station periodically needs a 'push' to remain in space. This push used to be performed by the space shuttle before it was retired: after the shuttle docked with the station for crew/supply transfers, it would typically fire it's engines to provide this push to keep the ISS in a stable orbit. Then it would detach and return to earth (using its thrusters for a retro-burn).
Part of the recent plans to decommission the ISS were due to there being no suitable tug to keep it in orbit after the retirement of the shuttle: one of my colleagues did his PhD on analysing how the ISS would burn up on re-entry.
So the only time a spacecraft needs to use its engines in space is for a) changing its orbit/flying to another planet, or b) overcoming atmospheric drag to prevent the orbit decaying.
The rest of the time, it is coasting with the engines off.
As one final point; space above 690km isn't entirely empty: solar particles and bits of dust are present throughout the entire solar system, out to beyond the orbit of pluto (see Heliosphere). Both Voyager 1 and 2 have left this region. So even comets will experience some (negligible) drag: I include this point that 'space' isn't completely empty for completeness.
There are a lot of good answers here already! But to sum it up:
Your spaceship DOES keep moving. It just doesn't move where you want it to.
In space, there is very little air or friction to slow anything down. So an object, whether it is a rock, a spaceship, or a planet - keeps going as fast as it had been going forever, although gravity might pull it around like it does anything else.
If your spaceship flies past the moon and then runs out of gas, it's "oops." The spaceship will just keep moving. The spaceship would need fuel to turn around, firing it's blasters in the opposite direction. (Gravity might be strong enough to pull it back, but that discussion is more detailed than we need here.)
There are exceptions: if you were in orbit around the earth, a very, very, tiny amount of air up there might slowly reduce the speed of your ship. Therefore, they will have to fire their thrusters once in a while to keep their speed (and altitude if you know about orbital mechanics) but they are not in fact using fuel all the time.
The earth is orbiting the sun. If your spaceship, far away from the earth, runs out of gas, it too will likely just orbit the sun for eons (or possibly escape the solar system.)
Seems two primary factors.
The man made space ships typically travel in very low altitude orbits, often where there is still significant atmosphere drag. The asteroids are in much larger orbits, usually about the sun, never coming too close to planets and traveling at much higher speeds. Consider voyager probes.
Additionally, asteroids and comments may be much more massive, more dense, and have a higher ballistic coefficient.
