I am really struggling with the following problem:
A car starts from rest with constant acceleration and moves in a straight line. At some point the car passes two marked road sections of length $L$ in the time $T$ and $\frac{T}{2}$. What is the acceleration, the velocity $v_L$ at the end of the first section and the total distance travelled until the end of the first section.
I am not sure if I am setting up the equations properly but here is what I got so far:
- I tried to draw a diagram
I calculated the average velocities and then used the equation $v^2-v_0^2=2a(x-x_o).$
$$ \implies \left( \frac{2L}{T} \right)^2-\left(\frac{L}{T} \right)^2=2a(L) \\ \iff \frac{3L}{2T^2}=a$$
It has the right units so I am somewhat confident that this is correct. The problem I have is using this acceleration to find the distance travelled since the start. The velocity at the end of the first section is $v_f=\frac{L}{T}$. I tried the following:
$$x(t)=\frac{1}{2}at^2 \space \space \space \text{and} \space \space \space v(t)=at \\ \implies v_f=\frac{3L}{2T^2}t \iff \frac{\frac{L}{T}}{\frac{3L}{2T^2}}=t \iff \color{red}{t= \frac{2}{3}T} $$
This can't be true because this equation says that it takes $\frac{2}{3}T$ to accelerate to velocity $v_f$ but from the question we know that the car needs time $T$ to travel a distance $$L+\text{distance travelled before first road section}$$
so I am getting a shorter time for a longer distance which can't be true. Are there any other equations I could use to relate distance and acceleration? Am I apporaching this problem and applying the formulas correctly?