Which approach can I use to find the total distance travelled by the car?

Question

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:

1. 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?

Answer 1

Let's say at the instant in time when the car encounters the first section the velocity is $v_0$, and velocity is $v_L$ when it passes the first section. The acceleration is a constant value $a$, and the car will travel a distance of $x_0$ in a time $t_0$ before hitting the first section.

This set up involves $5$ unknown values, so we need $5$ equations to solve this problem$^*$. They can all be your kinematic equations for the different parts of the journey:

The general equations to use are ones you are probably familiar with$^{**}$: $$\Delta x=v_i\Delta t+\frac12a(\Delta t)^2$$ $$v_f=v_i+a\Delta t$$

Applying these to each part we have:

$$x_0=\frac12at_0^2$$ $$v_0=at_0$$ $$L=v_0T+\frac12aT^2$$ $$v_L=v_0+aT$$ $$L=v_L\left(\frac T2\right)+\frac12a\left(\frac T2\right)^2$$

From here you need to solve the system of equations, which I leave to you.

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I can point out a flaw in your work though. You are assuming that the relation $v=\frac{\Delta x}{\Delta t}$ is true. This is not the case. That relation only holds for when $a=0$.

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$^*$Technically $L$ and $T$ are also "unknown", but they are "given" in the problem, so all of your answers will have to be in terms of $L$ and $T$.

$^{**}$ These are not the only equations you have to use. I see that you tried using $v_f^2-v_i^2=2a\Delta x$, but this equation can be derived using the ones I use above. So as long as you use two of the three general equations you should be fine.