MHB How far will the car travel in 10 seconds

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A car starting from rest accelerates at a constant rate of 2.0 m/s² for 10 seconds, resulting in a distance traveled of 100 meters. The calculation utilizes the kinematic equation d_f = d_i + v_iΔt + 0.5aΔt², confirming the final distance. The discussion also touches on using integrals to derive velocity and position equations, highlighting the relationship between them. Participants clarify the distinction between velocity (v) and position (x) in their calculations. The final consensus is that the car travels 100 meters in the given time.
karush
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$\tiny{Embry-Brittle \, 12}$
$\textsf{A car, starting from rest, accelerates in a straight line at a constant rate of $\displaystyle 2.0 \frac{m}{s^2}$}$$\textit{How far will the car travel in $10$ seconds}$\begin{align*}\displaystyle
\Delta t&=10\\
a&=2\\
d_i&=0\\
v_i&=0\\
d_f&= d_i + v_i\Delta t + \frac{1}{2} a \Delta t^2\\
&=0+0\cdot 10+\frac{1}{2} \cdot 2\cdot 10^2\\
&=\color{red}{100 \, m}
\end{align*}no answer given so hope this is ok :cool:
 
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karush said:
$\tiny{Embry-Brittle \, 12}$
$\textsf{A car, starting from rest, accelerates in a straight line at a constant rate of $\displaystyle 2.0 \frac{m}{s^2}$}$$\textit{How far will the car travel in $10$ seconds}$\begin{align*}\displaystyle
\Delta t&=10\\
a&=2\\
d_i&=0\\
v_i&=0\\
d_f&= d_i + v_i\Delta t + \frac{1}{2} a \Delta t^2\\
&=0+0\cdot 10+\frac{1}{2} \cdot 2\cdot 10^2\\
&=\color{red}{100 \, m}
\end{align*}no answer given so hope this is ok :cool:

$\displaystyle \begin{align*} v &= \int{ a\,\mathrm{d}t} \\ v &= a\,t + C_1 \\ v &= a\,t \textrm{ since the car starts from rest...} \\ x &= \int{ v\,\mathrm{d}t} \\ x &= \int{ a\,t \, \mathrm{d}t } \\ x &= \frac{1}{2}\,a\,t^2 + C_2 \textrm{ where } C_2 \textrm{ is the starting position...} \end{align*}$

So with $\displaystyle \begin{align*} t = 10 \textrm{ and } a = 2 \end{align*}$ we have

$\displaystyle \begin{align*} x &= \frac{1}{2} \cdot 2 \cdot 10^2 + C_2 \\ &= 100 + C_2 \end{align*}$

Thus the car has traveled 100 metres.
 
ok
I didn't know you could us a integral on it:cool:why did you shift from $v=$ to $x=$
 
Last edited:
You could use the kinematic equation:

$$x=\frac{v_f^2-v_i^2}{2a}=\frac{\left(20\frac{\text{m}}{\text{s}}\right)^2-\left(0\frac{\text{m}}{\text{s}}\right)^2}{2\left(2\frac{\text{m}}{\text{s}^2}\right)}=100\text{ m}$$
 
karush said:
why did you shift from $v=$ to $x=$

$v$ is velocity, $x$ is position.
 
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