# Why acceleration is limited?

In a frictionless environment, why doesn't an object move at infinite acceleration if force is applied on it?

Force causes movement, so unless there is an opposing force there shouldn't be any reason for the force to cause infinite acceleration.

• force causes change in movement. force doesn't cause movement – Ashwin Balaji Apr 28 '20 at 2:08
• You're confusing zero friction with zero inertia. – J.G. Apr 28 '20 at 9:40

Newton's Second Law states:

$$F=ma$$

So, for a finite (total) force, you get a finite acceleration. This doesn't impose a limit on your velocity, though. The longer you apply the force, the more your velocity increases. In classical mechanics, applying a force for an infinite time will result in infinite velocity. But classical mechanics doesn't hold at very high velocities. Instead, special relativity applies, which has a different relationship between force and acceleration that leads to the object's speed never passing the speed of light.

Because in a no friction environment an object obeys Newton's second law when a force is applied to it,

$$F=ma \tag{1},$$

the acceleration is decided by the mass of the object. When you apply a force to an object its mass decides how much it "resists" being moved by the force.

Newton's second law states that $$F=ma$$ for an object with constant mass. Unless the mass is zero, a finite force gives rise to a finite acceleration. If the mass is zero, then special relativity predicts that the object will travel at the speed of light.

Force does not cause changes in velocity; it causes changes in momentum. In Newtonian mechanics, momentum is proportional to velocity. In relativistic mechanics, it's not.

With light you could apply a force on a body, see radiation pressure on Wikipedia.

No other particles respectively bodies are moving faster than light, at least we haven't found any other.

Even if you can run at no more than 30 km/h, you can throw a ball with your arms at a higher speed. But such an addition of speeds is impossible for photons. Photons can neither push nor accelerate other photons, light cannot move faster than 300,000 km/h.

You might be confounding velocity and acceleration. If a particle is in a vacuum, then you have these scenarios:
1 - There's no force acting on the particle, or $$\frac{d}{dt}\vec v=\vec 0$$ which implies that $$\vec v$$ is constant (in direction and magnitude). Reminder: $$\vec 0$$ is also a constant vector, so the particle could be moving uniformly, or just standing in its place.

2 - There's constant net force acting on the particle, or $$\frac{d}{dt}\vec v=\vec a$$ where $$\vec a$$ is a constant vector (in both magnitude and direction). This means that velocity is going to increase infinitely since it is given by $$\vec v=\vec at+\vec v_0$$. You can see that $$v\to\infty$$ as $$t\to\infty$$.

3 - There's a variable (in either direction, magnitude, or both) net force acting on the particle, or $$\frac{d}{dt}\vec a=\vec a(t)$$. In this case, having the acceleration going to infinity depends on the equation that it is described by.
For example, if $$\vec a(t)=(t^3, t^2,t)$$, then it does indeed go to """$$\vec\infty$$""". However, if $$\vec a(t)=(\cos(\omega t),\sin(\omega t),0)$$, then although it is variable in direction, it's magnitude will always be the same.

As pointed out by other members, force describes the change in velocity (I assume constant mass), not acceleration.