Velocity

In its most simple form, velocity, or speed is the measurement of distance travelled over a time. Classically this is referred to as:
Speed = Distance / Time
More modern mathematics show it as:
Velocity (v) = Distance (s) / Time (t)
v = s / t
And you might even see:
Δv = Δs / Δt

The change in something, in maths is represented by, and therefore known by, the Greek symbol delta. It can be represented by the triangular Greek letter Δ or by an italicised d. Racing drivers have an interest in their lap speed but they also care a lot about improving it. They therefore want to see the changes in lap time, they are interested in the lap deltas. Additionally, the change in the delta is the gamma, which is your change in your change in velocity. Delta and gamma can be applied to anything, such as lap times, but an easier example would be a car with the driver having their foot to the floor on the acceleration pedal. Their staring velocity would be 0 and in 5 seconds they might be doing 50km/h (about 14m/s) and in 10 seconds they might be doing 80km/h (about 22m/s). Over the first 5 seconds their acceleration is their change in velocity so their acceleration is their delta (about 14m/s/s). As they go faster they go through more air per second so air resistance, or drag increases, which hampers the acceleration effort. Over the latter 5 seconds the cars velocity increases but the acceleration drops (to about 8m/s/s). The reduction in acceleration is the gamma (-6m/s/s/s). This information is handy as you can now ball-park the terminal velocity of the car, assuming the engine is up to it and in simplifying drag as a linier increase. Should you wish to work it out, it is about 25m/s or 90km/h. You could improve this calculation by laying a normal curve over the three data points but the best result would be to actually go out and drive the car to the max.

Acceleration is mathematically referred to as a derivative of velocity, as would be the gamma of the delta. Delta uses the capital Greek letter Δ so you can use the capital for gamma Γ, although it doesn’t read very well by eye in my opinion.
Gamma is just a Greek letter. Scientists, like mathematicians, also like to use Greek letters although sometimes for different representations. Nuclear scientists use alpha and delta to denote forms of energy irradiated in nuclear action. The lower case letter is used and its symbol is γ. I will only use this symbol to indicate gamma radiation.

In writing sci-fi I want specific velocities and times to add realism. Unfortunately, like the rest of rocket science, velocity in space is complicated as we need to deal with things like orbit circumference and time. Also we can’t talk about things like top speed in the same way we talk about cars. There is no air resistance and there is an infinite track so it gets a little silly at times.

Average Velocity

Velocity in itself is complicated, as sometimes it doesn’t seem to make sense. If you are traveling at a constant speed covering 6m in 2 seconds where v=s/t so you must be traveling a velocity of 3m/s. Now if you are instead at rest (not moving) at the start but still cover 6m in 2 seconds, then by the same calculation, you must be doing 3m/s as nothing has changed. But if you were accelerating, then at all times before the end you must have been doing less than 3m/s. If for the whole time except the very last moment you are doing less than 3m/s, then how could you have covered 6m in 2 seconds? It doesn’t make sense. The answer is because across 6m, you averaged 3m/s for 2 seconds. On a constant acceleration, any lack of speed at the start is made up at the end, meaning that the average lands in the middle. You must have been traveling at 3m/s after 1 second and your final velocity was therefore double it, 6m/s. Now it makes sense and the math doesn’t lie, but to understand you must comprehend what you are really asking. It’s the averaging of this element that leads to a lot of acceleration formula and derivative formula to having a 2 in it. If you are curious, the acceleration was 3m/s/s:
a = (v – u)/t
acceleration = (final velocity – start velocity) / time
Note: On the acceleration curve, you would have only travelled 1.5m in the first second.

Calculus – Derivation

In the previous example, if you asked ‘what speed did it travel’ and ‘what speed is it traveling’, you’d get two answers, which is not helpful. Knowing the exit speed from ‘what speed is it traveling’ is the better question as it helps with further questions. The question of ‘now how far will it travel in 10 seconds’ will require you to know it was traveling at 6m/s. In the time of Newton and Leibniz, around 1670, this issue of averaging providing the correct speed over a measured distance verses the exit speed, among other similar uses, was well known. It led to the next mathematical leap so it is quite logical that both Gottfried Wilhelm Leibniz and Isaac Newton both independently developed Calculus, as it is now known. Being a maths thing, they both got the same result, because being maths, you either got the same result or you didn’t get the result. The idea in a nutshell is that across an infinitely small distance, the average speed is also the final speed, so the maths can provide the same correct and absolute answer to both questions. Velocity became classified as a vector derived from distance and time. It’s handy to know what all this is about and why, so please research it, but I’m leaving it here in my attempt at an explanation.

Momentum

Momentum is implied by Newton’s laws of motion but specifically I’m interested in the energy it contains and its conservation as a vector quantity. It is a force and is measured in newtons but you only realise it when one thing hits another. It relates more to power but it is here as it is taken in part from velocity. The formula is:
momentum = mass * velocity
p = mv

Orbital Speed

Two bodies that orbit each other will exert a gravitational attraction to each other, so the Earth and the Moon orbit each other, which can be thought of as orbiting their centre of gravity. However, in regards to fictional example bodies A and B, if A is a star then the relative mass of B is almost irrelevant for anything less massive than another star. Also, if B is a spaceship or smaller, then its relative mass is irrelevant to anything Moon size or larger. In other words, for spaceships orbiting anything, then you only need to consider the mass of the planet or star that it is orbiting. The result is that the maths is simplified. If you do this with the Earth and the Sun you get an orbit time of 365.2 days, which is correct.

So to calculate it you first need a few things. These are:

G. This is big G. It is the universal gravitational constant and is needed to get the right answer. In a way it’s a measurement of the universe of profound significance as it needs to be factored in when we do the math. It is the difference between math without it and reality. I guess that’s why scientists call it Big G, even though the number itself is very small.
G = 6.67384*(10^-11)
G = 0.0000000000667384
The Suns mass rounded:
1.989*10³0kg
1,989,000,000,000,000,000,000,000,000,000kg
If you’re interested, Earth’s mass is
5.972*10²4kg
5,972,000,000,000,000,000,000,000kg
If you put the same number of digits in as for the sun you get:
0,000,005,972,000,000,000,000,000,000,000
At this point you may notice that the Sun is so massive that we actually lose the mass of the Earth in rounding! Equally we lose mass of the international space station in rounding of the Earth.
Earth’s distance from the sun. Remember that this has to be in metres and not km. The distance is a rounded average of the elliptical orbit.
Earth Distance from Sun = 149,600,000,000 metres.
where r is the radius (distance from the sun)
Circumference = 2 π r.
π is a lower case Greek letter pi. It represents another universe constant relating the amount of times a radius of a circle can be measured out on the circumference of a circle.
Earth’s orbit distance = 939,964,521,954m
There are 86,400 seconds in a day (60*60*24)

Simplifying for only one mass, orbital speed of Earth around the sun is given by the following equation:
√ ((G * Mass of Sun)/Distance from Sun)
The result is in seconds so you can divide it by 86400 to get days.

Asteroid Impact Speed

Ignoring atmospheric drag that limits sky divers to 54m/s, what we are talking about here is...

The Pull of Gravity. Little g

Rather than an asteroid, I’d rather talk about how fast we fall at the Earth’s surface. As the pull is constant and adds to any existing velocity, we are actually talking about acceleration. Earth’s pull at the surface is taught in schools and is 9.80665 metres per second per second. You might want to calculate the pull of gravity on a planet or around a star, so you’re going to need the following equation. Note that you will also need to know the distance from the centre of gravity, which for us is the radius of the Earth and it is:
6,371,000m

The equation, simplified for one mass is:
=G*(Earth Mass/(Earth Radius²))

It’s called little g as it only matters on Earth’s surface and nowhere else, so to physicists it’s a little boring and they are the ones who named it. In Sci-Fi it mostly relates to human interaction, of what feels normal. It matters that we evolved to expect 1g from the ground and that is what our bodies work best with. In that way it is exceptionally important.

Apparent gravity only occurs when the thing you’re in (relative to) is changing speed and pushing your velocity up. It’s this push that feels like the pull of gravity. So anyone in a spaceship traveling at a continuous speed will be apparently weightless in the direction of travel. Equally, a planet might pull you towards it accelerating you, but it will also be pulling your ship with equal force so although your velocity is accelerating relative to the planet, you are not accelerating relative to the spaceship. It is falling with you. In a nutshell, you need the ship you’re in to be accelerating at 9.8m/s to feel apparent gravity like on Earth. To be extra clear, apparent gravity isn’t gravity, just the illusion of it.
Orbit calculations are very specific so achieving orbit requires precise calculations. Knowing this, it may seem incredible that a planet has a moon orbiting it, but that is back-to-front logic. At a previous time anything that didn’t achieve obit either fell to Earth or flew away and now it is not there.
The gravity of something effectively remains constant but is in essence diluted over volume. Like light density it follows the inverse square rule. As you get closer to a mass, the pull of gravity gets bigger and the velocity required to escape (your escape velocity) increases.

Time to Target

As mentioned previously
Velocity = Distance / Time
So the time to target actually is distance divided by speed. I am going to go into more detail however as and when I refer to time to target, I am generally talking about straight line point to point, but in space there are three main versions of this:

Ballistic: No thrust, just something fired and left to keep going at the same speed. This is easy as it is just Time = Distance / Speed and my velocity interest ends here. It will be used when talking about BBs.
Missile: Your missile has an engine, maybe a rocket engine and it puts out a constant thrust which gives you a constant acceleration. You want your missile to take as short an amount of time as possible and hit with the highest speed possible. You have constant acceleration in one direction. I recommend you use the Astry Missile.
Zero Relative: You aim to arrive at your destination at a zero relative speed. Typically this would be for docking two ships together. For simplicity I will for now assume you will arrive at the same speed you started at. For the shortest amount of time you would burn your engines at max for half the distance accelerating up your speed, then ‘turn over’ where you flip your ship around and burn in the opposite direction to remove the velocity you added. Thors Hammer will use this.

v = s / t
velocity = distance / time
v = u + at
velocity = start velocity + (acceleration * time)
t = (v – u) / a
Time = (velocity – start velocity) / acceleration
A very handy rearrangement of the previous and worth remembering as it is easy to find yourself seeking more complicated equations to a calculation you need.
s = ut + 1/2at²
Distance = (start speed * time) + ((acceleration * (time ²))/2)
v² = u² + 2as or
v = √(u² + 2as)
velocity = √ (start velocity² + (2 * acceleration * distance))

Equation 5 is used to calculate missile velocity, however zero relative, using a turnover is not much more complicated. Its top velocity is a missile half the distance, and time wise you just double it to represent the same again but in the opposite direction (turn over).

Notes:
I at one point calculated that it wouldn’t take long to travel from one star system to another. Out of curiosity I worked out the top speed. Later, for something else, I looked at the speed of light. I then realised that my math was sound, except that it didn’t have the ultimate speed limit of the speed of light. My calculations were junk. So please look at your top speeds to make sure you remain within the realms of physics as near light speed you will have to account for relativistic effects.

Time Dilation

The speed of light is:
c = 299,792,458 meters per second
If your calculations put your velocity above this then you are in fantasy land. As you approach the speed of light you will experience time dilation, where one second for you will take longer to happen as seen by an outside observer.
=1/( √ (1-(v²/c²)))

You can measure speeds as a percentage of light speed, which can be handy as relativistic effects follow an exponential curve. The lower the speed the less the light speed limit bothers you. At 75% light speed it will cost you 1.5 seconds relative to every 1 second you experience. Even at 99% light speed you’ll only lose 7 seconds for every 1. It’s that last 1% that tends to infinity, and you really don’t want an infinite amount of time to go by in a single second for you. On the whole though, time dilation isn’t actually all that much to worry about.

Note that this section on time dilation is in regards to the special relativistic effects of velocity. Be careful of general relativistic time dilation effects of massive gravity wells like black holes.