# Question #2052e

Apr 6, 2016

If the girl is accelerating at $1.14 \frac{m}{s} ^ 2$, it will take her about $7.29 s$ to increase her speed by $8.32 \frac{m}{s}$.

#### Explanation:

First we’ll show the math, a derivation of the solution (in 6 steps). Then, we’ll explain each step in more detail.

1) ${v}_{f} = {v}_{i} + a \cdot t$

2) ${v}_{f} - {v}_{i} = a \cdot t$

3) $\frac{{v}_{f} - {v}_{i}}{a} = t$

4) $\frac{\Delta v}{a} = t$

5) $\frac{8.32 \frac{m}{s}}{1.14 \frac{m}{s} ^ 2} = t$

6) $t = 7.29 s$ ...and we're Done!!!!

1) Why did we choose this formula? Well, the second sentence asks a question containing the key phrase "increase ....speed". Ask yourself, what equations involve changing speed (due to a force or acceleration). There are two

${v}_{f} = {v}_{i} + a \cdot t$

and

${v}_{f}^{2} = {v}_{i}^{2} + 2 \cdot a \cdot d$

The 1st equation describes the change in speed when an object (or girl in this case) experiences an acceleration, $a$, for a length of time, $t$.

The 2nd equation describes the change in speed when an object experiences an acceleration, $a$, as it travels over a distance, $d$.

We chose the first equation here because the phrase "how long" refers to length of time!

What do the physical quantities (variables) in the equation we've chosen represent?

${v}_{i}$ is initial speed (or velocity)
${v}_{f}$ is final speed (or velocity)
$a$ is acceleration (the rate at which an object changes speed)
$t$ the amount of time the object experiences acceleration

2) In the 2nd step, we're just subtracting ${v}_{i}$ from both sides

3) Here, we're just dividing both sides by $a$

4) Uh oh! we don't know the initial and velocities ${v}_{i}$ and ${v}_{f}$! But that's ok! We only need to know the "increase in her speed", which is the DIFFERENCE between them for this problem. We denote this difference as $\Delta v$ (pronounced "delta v" or the "change in $v$"). In other words, $\Delta v = {v}_{f} - {v}_{i}$.

5) They tell us in the problem that the girl's acceleration $a = 1.14 \frac{m}{s} ^ 2$ and her increase in speed $\Delta v = 8.32 \frac{m}{s}$. So we replace the variables with their corresponding values.

6) The last step is arithmetic. We just divide to find $t$! So $t$ is how long it takes her to increase her speed by $8.32$!