Question

Why is acceleration inversely proportional to time squared?

Answer 1

Please refer to explanation

Explanation:

The S.I. unit of acceleration is `ms^(-2)` , but why?
Lets checkout.
we know the S.I. unit of distance is `m`.
and of time is `sec or s`
& velocity is displacement per unit time so unit of velcity become `ms^(-1)`

and acceleration is change is velocity per unit time
`a=(v-u)/t`
representing is S.I. units
S.I unit of acceleration`-> (ms^(-1))/s -> ms^(-2) or m/s^(2)`
[remember change in velocity will also have unit `ms^(-1)`].
hope it helps!!

Answer 2

See explanation

Explanation:

Its all to do with rate of increase in velocity

Suppose the velocity increased by 2 seconds at every second

Time in seconds`color(white)("ddd") 0,color(white)("d")1,color(white)("d")2,color(white)("d")3,color(white)("dd")4,color(white)("dd")5, ....`
`color(white)("dddddddddddddd")uarr color(white)("d")uarr color(white)("d")uarr color(white)(".")uarr color(white)("d.") uarr color(white)("d.")uarr`
Velocity in `color(white)("ddddddd")0,color(white)("d")2,color(white)("d")4,color(white)("d")6,color(white)("dd")8,color(white)("d")10, ....`
metres per second

Given that `ul("each increase in velocity")` is

`2color(white)(.) ("metres")/("second") larr " 2m per second" -> 2m/s`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("This is what can go wrong")`

The units of measurement fail `ul("in this approach")` so something is wrong with it:

Set time as represented by `t" seconds"`

Then the velocity at any time time `t` is:

#->2color(white)(.) ("metres")/(cancel("second")) xxtcolor(white)(".") cancel("seconds") = 2t" metres"color(red)(larr" Fail"#

`color(blue)("Making it work")`

We end up with only 'metres' as the unit of measurement. For velocity we must have metres per second `->("metres")/("second")`

To `ul("'FORCE'")` the above to work we must `color(red)("square")` the seconds.

So we use `color(white)(...)2color(white)(.) ("metres")/("second"^(color(red)(2)))color(white)(.)` instead giving:

`color(white)()`

#2color(white)(.) ("metres")/("second"^(color(red)(cancel(color(white)(.)2)))) xxtcolor(white)(".") cancel("seconds") = 2t("metres")/("second") ->2tcolor(white)(.) "m/s"#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Foot note")`

Another way of writing `color(white)("..")2 color(white)(.) ("metres")/("second"^(2)) ->2 m/s^2color(white)(.)` is `color(white)(.)2ms^(-2)`