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If in space with no opposing force,will a body of 1 kg pushed with 1 N force keep on accelerating 1 m/sec every second? When will it stop accelerating?

Yashvik gupta
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2 Answers2

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According to special relativity the equation of motion for an object under constant proper acceleration $\alpha$ is

$$\alpha=\gamma(v)^3a=\frac{1}{\sqrt{1-v^2/c^2}^3}\frac{dv}{dt},$$

where $a$ is the acceleration that you observe from an inertial frame and $\alpha$ is what you feel when you push down the gas pedal. In your case $\alpha=1\frac{m}{s^2}$.

The solution of that equation is $$v(t)=\frac{1}{\sqrt{\frac{1}{(\alpha t-v_0\gamma(v_0))^2}+\frac{1}{c^2}}}.$$

Therefore the velocity $v$ will approach lightspeed $c$, meaning that for any observer in an inertial frame it will keep getting faster but it will never reach $c$. The acceleration $a$ will keep getting smaller but never actually reach zero.

However for an observer which is accelerating together with your object, the acceleration will always be $\alpha=1\frac{m}{s^2}$. So a person in a spaceship can keep accelerating forever at a constant acceleration and he will notice that because he would feel a fictitious force proportional to $\alpha$ inside the spaceship.

Azzinoth
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According to newtonian mechanics, the body doesn't stop accelerating until you remove the force.
But in reality (According to special relativity)for an inertial observer yhe mass of an object moving with velocity $v$ varies as $$m =\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}.$$ Where $m_0$ is the rest mass of the object. As $v$ approaches to $c$ , the speed of light, the mass goes to infinity. Here $$F=ma.$$ If you keep the Force a constant (1N), then the acceleration approaches to 0 as time varies (since increase in speed of object to $c$ tends the mass to infinity). Thus as soon as object approaches the speed of light it stops accelerating so that no objects exceeds the speed of light.

walber97
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