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

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?

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.

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 $$a=0$$.

$$^*$$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.

• Thank you so much for that great answer.I think I am going to apply this "method" of checking how many unknowns I have and then deducing how many equations I need to get a solution a lot more in future problems. I always get confused when I have a lot of unknowns and I think the way you wrote it down is a great way to keep it organized. I see my mistake in assuming "v=\frac{\Delta x}{\Delta t}" is true. I am going to work on it tomorrow (already 1 am at my place) and hopefully get a solution. Thank you again. – Nullspace Nov 4 '18 at 0:04