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How Fast Would You Have Have To Be Going In Order To Out Run A Rocket Car Traveling This Fast

If I run forward in a spaceship which travels at the speed of light, will my speed exceed the speed of light?

Being composed of matter, neither you nor your spaceship can reach the speed of light.But in principle, the ship could be going at one mile per hour less than the speed of light, and you could run faster than than 1 mile per hour. You run in the direction the ship is moving. So you’d be going faster than the speed of light?Wrong.Let’s unpack it.I’ll choose to consider myself at rest, and observe you and the spaceship going insanely fast. I observe your clock on the wall of your ship and it is running slow, due to time dilation.You will see nothing odd about your clock, and measure your jog at 3 mph.However, your clock is running so slow relative to mine, the time that you require to cover a given distance between two marks on the spaceship will be more than measured on your clock. Not only that, but if I hold up a tape measure to those marks as they flash by, my tape will give a smaller distance than yours will. So you are taking more time to cover less distance inside the spaceship.So it’s not surprising that relative to me you are still going at less than the speed of light. And a bacterium swimming through you may be going faster than you, but still will be going at less than the speed of light. And a molecule inside the bacterium, and an atom inside the molecule, and an electron inside the atom, and so forth.No matter how many velocities you combine you still won’t reach the speed of light.Adding velocities only works for things that are going significantly slower than the speed of light.The correct rule is called the composition of velocities. It is approximated by adding them if they are slow velocities.Velocity-addition formula - Wikipedia

Why can't astronauts just take off from a runway instead of the rocket launch pad?

In theory they could. But no airplane invented yet can go even 1/5 as fast as orbital speed, and if you are going slower then you start falling down (or gliding down) as soon as your fuel runs out.

A rocket has such powerful engines it can accelerate straight up, getting to space in under 2 minutes, where it then turns sideways and spends the next 8 minutes getting up to 17,500 MPH orbital speed. Such a rocket is more efficient in a vacuum and anyway no plane or rocket could go that fast in the air without burning up from friction with the air. Also, rocket engines need a ridiculous amount of fuel and burn it ridiculously fast, so rockets are really heavy with fuel at takeoff. And they are BIG, with most of the size being the fuel tank. Look how big that orange fuel tank is that the Shuttle has! No wheels or wings ever invented are strong enough to allow such a heavy vehicle to roll down a runway and fly like a plane.

Would it be possible for a solar car to travel faster than the speed of light?

It is impossible *as far as we know*, which doesn't rule it out in the long run. Some scientists claimed to have gotten around the limit already:
Mostly, those who postulate the existence of particles that travel faster than the speed of light, also admit that it is impossible for said particles to cross the speed of light boundry.

However, to explain the current rule, you can look to Einstein's theory of relativity. There are many equations, but they basically say that as velocity approaches the speed of light, your mass would increase to infinity, your length would decrease to infinity, and time would dialate to infinity (stand still). We cannot mathematically explain anything crossing the speed of light boundry. To be more precise than saying "infinity", what actually happens is that a term in the equation becomes a complex number, causing a singularity.

Traditional methods of propulsion (rocket engines, jets, ion engines, solar sails, etc) are of no use either. This is becasue that the amount of energy necessary to increase one's velocity increases as you approach the speed of light. When you are very close to the speed of light, in order to speed up more, you would need vast amounts of power. We just don't have the technology to do it that way.

An experimental rocket car starting from rest reaches a speed of 560km/h after a str8 400m run on a level -?

You mush use the equations of straight-line motion.

x= distance: 400
v= final velocity: 155.55 m/s
u= initial velocity : 0
t= time : ?
a= acceleration: ?

v^2= u^2 + 2ax
155.55^2 = 0 + 800a
a = 30.24 m/s/s

v = u + at
155.55 = 30.24t
t = 5.14s

Is it possible for a car going 130 mph to run a 1/4 mile in 5 seconds?

A car travelling at 130 mph covers 0.18 mile in 5 seconds, which falls short of the 1/4 or 0.25 mile. Therefore the answer to your question is no.

Or, if you question is if a car can accelerate in 5 seconds to cover a 1/4 mile, what is it velocity as it reaches the 1/4 mile?

The answer to that question is 360 mph. No production car can achieve that kind of acceleration or have that speed. However there are "rocket cars" that can do that.

Why do rockets need so much fuel if, in space, there is nothing to slow them down once they are up to speed?

The question makes it appear as though “once they are up to speed” was not a big deal.But it is. It is a huge deal. To get into the lowest possible orbit around the Earth, a spacecraft must accelerate to 7.8 km/s. Which is around 25 times the speed of sound.Worse yet, unlike airplanes, rockets cannot use air-breathing engines. And that’s a really huge deal. For instance, if you burn hydrogen, every two atoms of hydrogen require an atom of oxygen to burn (and produce H2O, aka. water.) The atomic weight of hydrogen is 1; the atomic weight of oxygen is 16. So for every 2 kilograms of hydrogen, you need to carry 16 kilograms of oxygen. In other words, if the rocket fuel is liquid hydrogen, we just increased the amount of fuel that a rocket must carry by a factor of 9, as it also has to include the oxidizer.And all that fuel has to be accelerated along with the rocket. Well, all of it other than the fuel that has already been used, so less and less force is required as the rocket gets lighter (this is why multistage rockets are used; when the rocket is too light, running a huge engine at low power would be quite inefficient or maybe not even possible, so it is better to just lose all that dead weight and continue with a smaller engine in the second stage.)In short, almost all that rocket fuel is used up in the first few minutes of flight, lifting the spacecraft to an altitude of several hundred kilometers and, most importantly, accelerating the spacecraft to (at least) 7.8 km/s.For spacecraft that are going to deep space, a fair bit of additional fuel is required to accelerate to 11 km/s (the escape velocity of the Earth) and then some, to achieve the desired orbit around the Sun that will allow them to intercept their intended target (maybe another planet) months or years later; and then there is more fuel needed to decelerate at the target, so that they can enter orbit around the target. This “delta-vee” (additional change in velocity) requirement could be several km/s. But these fuel requirements are still minuscule compared to the initial fuel load that is needed to get into orbit in the first place. Other spacecraft, which orbit the Earth, may require very little fuel “once they are up to speed”; this fuel will be used to adjust their orbit if necessary, or adjust their attitude.

Why can't a rocket gradually fly away from the Earth at 10 mph?

In 1979 there was a TV series called Salvage 1, in which Andy Griffith played a junkyard operator who built himself a space ship out of spare parts, and used it to travel to such places as the moon to gather up scrap material from old space missions, which he’d bring back and sell. The thing is, his space ship couldn’t go very fast. Instead, he had discovered some kind of unique propulsion system (if I remember right) that continuously pushed the ship just enough to keep it chugging along at some fairly slow speed — all the way to the moon.And in theory, this idea would work. As long as you’ve got enough fuel to continually produce a thrust that exceeds the pull of the earth’s gravity, you can get arbitrarily far away from the earth at an arbitrarily slow speed.But for chemically fueled rockets, this is a terribly inefficient way to use your fuel if you want to get very far from earth. It’s much more fuel-efficient to quickly reach (or exceed) escape velocity, and then shut off your engines. Once you have reached escape velocity, your momentum alone is enough to carry you arbitrarily far away from the earth.* Basically, you get to coast the rest of the way (except for mid-course corrections), until you get near your destination.In recent years, we’ve been perfecting “ion propulsion” engines, which work on a different principle than chemical rockets. They produce a very small thrust (too small to lift a space ship off the ground; they’re only useful once the vehicle is already in space), but they may be able to run continuously for years. Ion propulsion systems may someday overturn the traditional “jump to escape velocity” method of interplanetary travel.*Note: I know I said you can get “arbitrarily far” from the earth if you reach earth’s escape velocity; well, that’s not entirely true, because there’s this other big, monstrous thing also pulling on you — the sun. Merely reaching earth’s escape velocity (or a little above it) will actually just put you into orbit around the sun. You’ll have to reach the sun’s escape velocity — as a few vehicles like the Pioneer and Voyager probes have — if you intend to go really far from earth.