# 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?

Apr 6, 2018

$7.5 \frac{m}{s} ^ 2$
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:

$\textcolor{b l u e}{\text{The teaching bit}}$

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 $\to 2 \frac{m}{s}$)
The next second your speed has increased by a further 2 m/s. So you are travelling as $\left(2 + 2\right) \frac{m}{s} \to 4 \frac{m}{s}$

Next you have $\textcolor{w h i t e}{\text{ddd.d}} \left(4 + 2\right) \frac{m}{s} \to 6 \frac{m}{s}$

Next you have $\textcolor{w h i t e}{\text{ddddd}} \left(6 + 2\right) \frac{m}{s} \to 8 \frac{m}{s}$

And so on. So the increase is $2 \frac{m}{s}$ 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"))
$\textcolor{b l u e}{\text{metres for 1 second ")color(red)( "for every second}}$
..................................................
Lets consider the units of measurement (in brown)

velocity$=$acceleration x time

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dd")vcolor(white)("d.d")=color(white)("dddd") acolor(white)("dddd")xxcolor(white)("d}} t}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dd")v color(brown)(m/s)=color(white)("dddd")acolor(brown)(m/s^2)color(white)("dd")xxcolor(white)("d}} t \textcolor{b r o w n}{s}}$

Notice that the units for time can be cancelled out getting rid of the 'squared'. So we end up with just $\textcolor{b r o w n}{\frac{m}{s}}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dd")v color(brown)(m/s)=color(white)("dddd")acolor(brown)(m/s^cancel(2))color(white)("dd")xxcolor(white)("d}} t \textcolor{b r o w n}{\cancel{s}}}$

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$\textcolor{b l u e}{\text{Answering the question}}$

$a t = v$

$a \left(8\right) = 60$

$a = \frac{60}{8}$

$a = 7.5$

But this is in $\frac{m}{s} ^ 2$ so we have:

$7.5 \frac{m}{s} ^ 2$