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

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?

## marked as duplicate by JMac, Qmechanic♦Apr 10 at 16:39

• – JMac Apr 10 at 13:00

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.

• will it ever really reach the speed of light? – Yashvik gupta Apr 10 at 18:49
• @Yashvikgupta No, it can never reach the speed of light. – Azzinoth Apr 10 at 18:52

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.

• en.wikipedia.org/wiki/Mass_in_special_relativity#Controversy Relativistic mass isn't much used nowadays. – Gert Apr 10 at 13:41
• You could have a good result by saying $F=dp/dt$. Then you would not need to introduce relativistic mass, which is a misleading concept – Ballanzor Apr 10 at 15:52
• will it every really reach the speed of light? – Yashvik gupta Apr 10 at 18:49