# Distance and velocity increased by $g$ units every second

1. Distance fallen in every second gets increased by $$g$$ units.
2. Velocity is increased by $$g$$ units every second.

Is it true that distance and velocity increases by $$g$$ after every second?

If it starts from rest, then distance in the first second will be, by $$s=ut+1/2at^2$$, $$5 \;\text{m}$$. Then if it increases by $$g$$, will it be $$15 \;\text{m}$$? Similarly, the velocity, by $$v=u+at$$, will it be $$10 \;\text{m/s}$$ in the first second? Will it increase like $$20 \;\text{m/s}$$, $$30 \;\text{m/s}$$, $$40 \;\text{m/s}$$, ... every second?

P.S. $$g$$ is acceleration due to gravity, which I assume to be $$10 \;\text{m}\,\text{s}^{-2}$$.

Is it true that distance and velocity increases by $$g$$ after every second?

That's true for velocity. It's not true for distance, but that's not what your quoted statement says.

1. Distance fallen in every second gets increased by $$g$$ units.

That's talking about the additional distance covered in each second, not the total distance, which increases quadratically, that is, in proportion to $$t^2$$. Here's a table for the first 5 seconds, using $$g = 10 \,\mathrm{m/s}$$ and $$s = \frac12 gt^2$$

time distance increase
0 0
1 5 5
2 20 15
3 45 25
4 80 35
5 125 45

In each second, the amount of distance covered in that second is $$g$$ metres greater than the distance covered in the previous second.

FWIW, Galileo measured and described falling motion in this way, in terms of the additional distance covered per unit time.

• A table on an SE site... I never knew that was possible. Sep 9 at 11:27
• @Steeven We got Markdown tables late last year. See meta.stackexchange.com/q/356997/334566 Of course, on Physics.SE we can also use MathJax tables. Sep 9 at 11:30

An object falling has a constant acceleration $$g$$. This means at any time $$t$$ its velocity will be given by $$v=v_0+gt$$ If it has an initial velocity $$v_0=0$$ then $$v=gt$$ So its velocity will increase by a factor $$10$$ every second. In other words,

after one second it will have a velocity $$10ms^{-1}$$

and after two seconds it will have a velocity $$20ms^{-1}$$

and after three seconds it will have a velocity $$30ms^{-1}$$

and after $$t$$ seconds it will have a velocity $$v=10\times t \ ms^{-1}$$

So for every consecutive second the velocity has changed by $$10ms^{-1}$$. We say it has an acceleration of $$10$$ meters per second, per second or "$$10$$ meters per second squared".

As you have stated, the distance it falls will follow a different relationship given by $$x=v_0t+\frac{1}{2}gt^2$$ or $$x=\frac{1}{2}gt^2$$ if $$v_0=0$$. Then

after one second it will have fallen $$5m$$

and after two seconds it will have fallen $$20m$$ change=$$15m$$ $$\Delta$$change=$$10m$$

after three seconds it will have fallen $$45m$$ and change$$=25m$$ $$\Delta$$change=$$10m$$

after four seconds it will have fallen $$80m$$ and change$$=35m$$ $$\Delta$$change=$$10m$$

after five seconds it will have fallen $$125m$$ and change$$=45m$$ $$\Delta$$change=$$10m$$

and after $$t$$ seconds it will have travelled $$x=5\times t^2\ m$$ and in this instance, the difference between the distances after consecutive seconds will always be $$10m$$.

There is a quadratic relationship between $$x$$ and $$t$$ as oppose to the linear relationship between $$v$$ and $$t$$, but differences shown above after each second are always 10 units. So both of the statements

1. Distance fallen in every second gets increased by $$g$$ units.
2. Velocity is increased by $$g$$ units every second.

are true but the first statement talks about additional distance after each second.