If you have a rocket that constantly uses fuel, then you have a constant acceleration in the universe, right? Wouldn’t you eventually reach the speed of light?
Answer
To accelerate a mass, you must exert a force on it. The acceleration is then equal to the force divided by the mass. So if the mass is constant, a constant force will also give a constant acceleration (in the absence of resistance, friction…). A constant acceleration means a linearly increasing speed : you are indeed going faster and faster.
However, this reasoning only applies to speeds that are negligibly small compared to the speed of light. Then the above “Newtonian” reasoning fits perfectly.
The problem is that, at velocities that get very high, that is, a non-negligible fraction of the speed of light, the mass is no longer constant. At relativistic velocities, mass increases according to
m = mO / carrot [ 1 – (v/c)2 ]
where mO is the “ordinary, everyday” mass, the rest mass. You see that the square root in the denominator is quasi one at low, everyday speeds, so that this formula is then perfectly the Newtonian formula
F = mO.a becomes. However, as v increases and becomes comparable to c, the denominator becomes smaller and smaller, so that the mass m that you still want to accelerate becomes larger. If v in limit reaches the value of c, the denominator would become 0, so that m becomes infinitely large.
In order to reach the speed of light you would have to be able to accelerate an infinitely large mass in limit, and that is of course not possible because that would cost an infinite amount of energy. You will never reach the speed of light
see also : http://www.physlink.com/education/askexperts/ae161.cfm
Answered by
prof.dr. Paul Hellings
Department of Mathematics, Fac. IIW, KU Leuven
Old Market 13 3000 Leuven
https://www.kuleuven.be/
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