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You will need to advance your way through career mode. Best ship is completely dependent on your available parts and desired result. Sometimes you just need suborbital capabilities, sometimes low orbit, sometimes interplanetary craft. Each requires different design.
Watch some YT videos to see how various people design their crafts or simply go by trial and error.
At such an early stage, you need to gather as much science as possible, so build single stage crafts, go anywhere on Kerbin, gather soil samples, eva and crew reports etc, and recover this all back to the space Center. Once you have enough science, you can then begin to unlock better parts, and figure by way of trial and error to build the ships to do your missions. Missions also give you a direction to go with your building, so pick up the basic ones and try complete them.
Everyone to their own, just remember a parachute!
Or as RoofCat points out, there are Training missions, There are also Scenarios that can give you examples
http://steamcommunity.com/sharedfiles/filedetails/?id=899719043
...but sometimes they have to be complicated...
http://steamcommunity.com/sharedfiles/filedetails/?id=538317910
...but all they have to do in the end is get to space and do sciencey stuff.
As a gerneral rule for your launch stage, make sure the thrust to weight ratio (TWR) is high enough to get off Kerbin which has a gravitational acceleration of 9.8m/s^2.
You can find your TWR or acceleration by adding the force all the active engines measured in Kilonewtons (kN) and dividing the mass of your ship measured in kilotons (1000kg or 1Mg).
That fancy equation F=ma means your acceleration is equal to Force/Mass.
Get your acceleration over 10m/s^2 (or a TWR greater than 1) and you'll get off the ground.
The TWR changes depending on which planet you're close to so I prefer acceleration measurements.
Getting off the ground is neat but then you have to get through the atmosphere.
You'll need wings but wings add mass.
The best place to place wings is as far behind or below the center of mass as possible.
The trick is to add just enough wings to control your ship but not so many that it's too heavy to launch.
Adding struts helps a lot too by reducing or eliminating rocket wobble.
Autostrut is a newer stock feature enablable through the main menu options if you turn 'advanced tweakables' on.
It works like struts but doesn't add mass like struts do.
Actually flying through atmospheres and when to start your gravity turn is a topic for another thread.
Designing a rocket that can get to orbit is the tricky part.
You need to find out how much dV (total change in velocity) your ship has and that number has to be 3km/s ish.
There are 2 equations to find your dV:
ISP * 9.8 * LN(rocket wet mass / rocket dry mass)
This just takes your mass and engine ISP into account for calculating dV.
This one should work for your entire rocket but I never tried it for the whole rocket at once.
(Total engine thrust * total fuel) / ((wet mass + dry mass) / 2 * fuel used per second)
This is my personal favorite formula.
This one has to be done for each stage.
Slap enough dV on your rocket and you can go literally anywhere.
You can get that Kerbal Engineer mod to do all this for you if you want but I'll be more impressed if you don't.
Once you get to orbit, you're halfway to anywhere.
Make sure you have some way to control your attitude.
RCS works but adds mass and uses monopropellant and doesn't require electricity.
Reaction wheels let you rotate your stuff as much as you want as long as you have electricity.
Don't forget batteries and antennas if you plan on transmitting science data back to Kerbin.
If you have solar panels, put some on so that at least 1 has sunlight at any given time.
Most engines have an alternator in them so they'll generate electricity while they're on.
Landing somewhere is just like taking off but in reverse.
All the same construction stuff goes into a landing ship but you need to make sure the center of mass is as low as possible and the legs are as far apart as possible or you're going to tip over and won't be able to get back up if you land on a slight incline.
tldr;
More struts
More boosters
Don't forget your parachutes.
Ask and ye shall receive.
If you need help with anything, let me know. Even if you need help handing out happy meals XD
nice profile pic
i'd be willing to insert some screenshots if I kew how