# Acceleration - Constant force vs force for an instant

I am reading about Newton's second law, and I am slightly confused.

So I think I get that $F=ma$ can be used to find acceleration. I am struggling between two scenarios: constant force acting over a period of time and force acting for an instant.

Examples:

1. A constant force of 5N acts for 5 seconds on a one kilogram mass. I am assuming you can use $F=ma$ to find $a=$5 m/s2. So after 5 seconds, the velocity would be $v=u+at$.
2. What happens if a force of 5 N acts only for an instant, like a push to a ball at rest? How do you find acceleration? How do you find velocity after, say, 5 seconds?
• $F=ma$ doesn't really say anything about time the way you see it. All three, $F$, $m$ or $a$ may be (or may not be) functions of time. It just gives the relation between the three for each instant in time.
– Gert
Commented Nov 8, 2015 at 0:26

You might want to use the idea of impulse, $J$, defined as $$J=\int_{t_a}^{t_b} F\,\mathrm{d}t=\Delta p=mv_f-mv_0$$ In your first case, $F$ is not time dependent, and so you have $$J=Ft_b-Ft_a=mv_f-mv_0$$ You should be able to solve this.

In the second case, $F$ may or may not be time dependent. The equation for impulse can be changed to $$v_f=\frac{\int_{t_a}^{t_b} F\,\mathrm{d}t+mv_0}{m}$$ If $t_a=t_b$ - your second case - then $v_f=v_0$. Therefore, this proves that a force applied for an instantaneous amount of time produces an acceleration lasting for an instantaneous amount of time - which causes no change in velocity whatsoever.

1. A constant force of 5N acts for 5 seconds on a one kilogram mass. I am assuming you can use F=ma to find a=5 m/s2. So after 5 seconds, the velocity would be v=u+at.

Yes.

If a constant force $F$ acts on something for 5 seconds, then that something accelerates with acceleration $a$ during those five seconds.

In other words: That object accelerates with $a$ in the first instant, in the next instant, after 1 second, after 2 seconds etc. The point is, Newton's law has got nothing to do with time. A force accelerates something right away. For how long it acts doesn't change the acceleration - time just affects the final speed.

1. What happens if a force of 5 N acts only for an instant, like a push to a ball at rest? How do you find acceleration? How do you find velocity after, say, 5 seconds?

You find acceleration in the same way. The acceleration is again $5 \;\mathrm{m/s^2}$.

The velocity $v$, though, is found through the motion equation you just used before: $v=v_0+at$. If the time is infintely small, because the force only acted in an instant, then the acceleration $a$ also only happened for an instant. This will not result in a very large velocity - the smaller the time, the less the velocity.

What happens if a force of 5 N acts only for an instant, like a push to a ball at rest? How do you find acceleration? How do you find velocity after, say, 5 seconds?

If the 5 N force acts only for an instant (say $100 ms$) on that $1 kg$ mass the acceleration will be $5m/s^2$ for $100 ms$ and zero after that.

During those $100 ms$ the body will accelerate from $0 m/s$ (if it is at rest initially) to $5 m/s^2*100 ms = 0.5 m/s$ and after that the mass will maintain those 0.5 m/s indefinitely.

In conclusion, after $5 sec$, the $1 kg$ body will travel at a constant speed of 0.5 m/s.

• My answer is good. The minus I received is just to discredit me. Commented Nov 15, 2015 at 16:18