Question

If an object is moving at `0 m/s` and accelerates to `60 m/s` over 8 seconds, what was the object's rate of acceleration?

Answer 1

`7.5 m/s^2`
The actual calculation is very short (at the end) but I have also included some teaching (at the beginning) which is a bit long.

Explanation:

`color(blue)("The teaching bit")`

First lets understand what acceleration is:

At each second that passes you are going faster than the previous second. Have a look at this example:

At time 0 you are not moving. By the time the first second is reached you are travelling at 2 metres in 1 second ( 2m/s `->2 m/s`)
The next second your speed has increased by a further 2 m/s. So you are travelling as `(2+2) m/s ->4 m/s`

Next you have `color(white)("ddd.d")(4+2)m/s->6m/s`

Next you have `color(white)("ddddd")(6+2)m/s->8m/s`

And so on. So the increase is `2 m/s` for every second
Written as: #2 color(blue)(ubrace(color(white)("d")m/scolor(white)("d")) )color(white)("d")xxcolor(white)("d") color(red)(ubrace(color(white)("d")1/s color(white)("d")))color(white)("ddddd") ->color(white)("dddd")2ubrace(color(white)("d")m/s^2color(white)("d"))#
`color(blue)("metres for 1 second ")color(red)( "for every second")`
..................................................
Lets consider the units of measurement (in brown)

velocity`=`acceleration x time

`color(green)(color(white)("dd")vcolor(white)("d.d")=color(white)("dddd") acolor(white)("dddd")xxcolor(white)("d")t)`

`color(green)(color(white)("dd")v color(brown)(m/s)=color(white)("dddd")acolor(brown)(m/s^2)color(white)("dd")xxcolor(white)("d")tcolor(brown)(s))`

Notice that the units for time can be cancelled out getting rid of the 'squared'. So we end up with just `color(brown)(m/s)`

`color(green)(color(white)("dd")v color(brown)(m/s)=color(white)("dddd")acolor(brown)(m/s^cancel(2))color(white)("dd")xxcolor(white)("d")tcolor(brown)(cancel(s)))`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

`at=v`

`a(8)=60`

`a=60/8`

`a=7.5`

But this is in `m/s^2` so we have:

`7.5 m/s^2`