51 How far will the car travel in 10 sec

[karush]

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

 

[Kansas Boy]

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.

 

[karush]

not sure looks like 10 years ago
but I am still weak on physics stuff