MHB 51 How far will the car travel in 10 sec

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The discussion focuses on calculating the distance a car will travel in 10 seconds with an acceleration of 2 m/s², starting from rest. Using the formula for distance, the final calculation shows the car travels 100 meters in that time. Participants discuss the complexity of the calculations and the integration of acceleration to derive the distance formula. There is also mention of potential typos and a desire to create a graphical representation of the motion. Overall, the conversation highlights the mathematical approach to solving motion problems in physics.
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How far will the car travel in $10$ seconds"
\begin{align*}\displaystyle
\Delta t&=10,\quad a=2,\quad d_i=0,,\quad 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=100 \, m
\end{align*}

or $\displaystyle v=\int{ a\,\mathrm{d}t}=a\,t+C_1 =a\,t$
since the car starts from rest... $\displaystyle x=\int{ v\, dt=\int{ a\,t \,dt} = \dfrac{1}{2}\,a\,t^2 + C_2 $
where $ \, C_2 = 0 \displaystyle x=\frac{1}{2}\cdot 2\cdot 10^2+C_2= 100+0=100 \, m$

ok i am sure there are some typos
but isn't this more complicated than it need to be
I was just looking at some examples and took stabs at it
Mahalo Much

also I not asked for but wanted to try a tikz graph of this if it is correct
 
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I am not clear what your purpose is in posting this. It is true that if a vehicle is moving with acceleration a meters per second per second, with initial velocity $v_i$ meters per second then in t seconds it will have reached velocity $\int a dt= at+ v_1$ meters per second and will have moved distance $\int at+ v_1 dt= \frac{a}{2}t^2+ v_1t$.

Integrating acceleration with respect to time, twice, gives the formula you use first. Then, of course, doing the integrations gives the same thing.
 
not sure looks like 10 years ago
but I am still weak on physics stuff
 
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