Hello,
I am Francesc Soriguera
and I will present this video: one exercise on the physics of movement.
In this exercises we will determine
the safe space-gap between two consecutive vehicles.
This will require to apply the equations of the uniform accelerated motion to determine the stopping distance.
Then we will illustrate the solution in the time-space diagram.
I will finish the video presenting and Excel spreadsheet as further material to
play with the trajectories resulting from this problem.
So, let's go over the statement of the problem
Consider two vehicles following each other, we will call them the leader, l, and the follower, f,
which are travelling at constant speeds, v_l and v_f.
The braking capabilities of the two vehicles
yield maximum decelerations of a_l and a_f, respectively.
So, a_l, a_f < 0 are decelerations,
the are negative.
The problem asks
to determine, in terms of the variables given, the minimum space-gap between the vehicles
so that, if the leader suffers an incident and starts braking at full power,
the follower is able to reach a full stop, avoiding collision,
given a reaction time of ∆t (i.e. the follower vehicle starts braking ∆t units of time later than the leader).
the solution process will consiste in the following steps:
first will set the initial conditions and the system of references;
after we will compute the stopping position of the leader an of the follower;
and finally, we will impose the condition to avoid collision and determine the safe space-gap.
So, let's start by setting the system of references and initial conditions.
We consider that the leader starts braking at t=0.
We also consider that the position of the vehicles is determined by their rear bumper.
Then we consider that the position of the leader at t=0 is x_l(0)=0.
Given these conditions, the initial position of the follower
will be, at instant 0,
x_f(0) equals to minus g_f minus l_f,
where g_f is the space-gap left by the follower.
This is the distance between the front bumper of the follower and the rear bumper of the leader,
and l_f is the follower vehicle’s length.
You can see this in the picture below the slide
where you can see the initial conditions and the initial
space-gap between the vehicles.
Now, we need to compute the stopping position of the leader
and the stopping position of the follower.
For the leader, this is simply done by considering that this is an uniformly accelerated motion,
where the acceleration is a_l, the initial speed is v_l, and the initial position is 0.
Applying the equations for the position, for the stopping distance
of the uniformly accelerated motion,
we get this equation in the slide,
so that, the stopping distance, the stopping position of the leader is equal to minus
the speed of the leader squared, divided by two times
the acceleration of the leader. Note the negative sign, which means
that, because the acceleration is negative, the stopping position will be positive.
Then we compute the stopping position of the follower,
considering that it is a uniformly accelerated movement,
with an acceleration a_f, speed v_f,
and the important thing here is that the inicial position
of the follower at braking is not the time 0
it is time ∆t because it considers the reaction time.
The inicial position of the follower at the instant of braking
is minus l_f minus g_f, which is the position at time cero,
plus ∆t times v_f. This ∆t times v_f is the distance covered
by the follower during the reaction time traveling at v_f.
Then we can apply again the equations of the stopping position
of the uniformly accelerated motion and we obtain that x_f,stop is equal to minus v_f squared
divided by two a_f, minus l_f,
minus g_f, plus ∆t times v_f.
This gives us the stopping position of the follower.
Then the next step is to impose the condition to avoid collision.
this is that x_f,stop plus l_f
so, the stopping position of the follower plus the vehicle length,
needs to be smaller than the stopping position of the leader, as you can
see in the figure in the slide, so you can see that to avoid collision
this condition needs to be fulfilled.
Then we just substitute x_f,stop and x_l,stop, by the equations that we derived in the previous
slide and you obtain this condition here.
Then, just by solving for
the minimum safe space-gap, we obtain the solution
of the problem. This is that the minimum safe space-gap
it needs to be larger or equal to the speed of the leader squared divided by two times de acceleration
of the leader minus the speed of the follower squared divided by two
times the acceleration of the follower plus ∆t times v_f.
This is the solution of the problem that we were looking in this exercise.
Note that there is particular case
when both vehicles travel at the same speed
and have identical braking capabilities. So, if both vehicles,
the leader and the follower, travel at v0 and both can decelerate at
a maximum deceleration a_b,
then the minimum safe space-gap, g_f,
is simply the ∆t times v_0.
Note that only depend on reaction time and initial speed,
because the braking distances of both vehicles
will by exactly the same.
Now, we are going to represent this solution in the time-space
diagram and I will illustrate the particular case where
the deceleration of the two vehicles is equal to - 6 m/s^2,
the initial speed of the vehicles is 30 m/s,
the reaction time is 1 s, and the length of the vehicles is 5 m.
Then applying the solution that we have obtained, we obtain that the safe space-gap
is 30 m. If we plot this
into a time-space diagram we obtain these plots.
In the top left of the slide you can see the vehicles' trajectories, so the quadratic trajectories,
the quadratic deceleration
trajectories, of the vehicles,
of the leader and the follower, and you can see how they decelerate until a full-stop,
and they reach an horizontal trajectory,
where the vehicles are stopped without colliding. Below,
at the left hand side, you can see evolution of the space gap,
how its reduced from the minimum
safe-space gap of 30 m,
until zero when the vehicles are stopped. Note that we have left
no margin of error here. In reality, we should have left a bit
of margin of errors, so that the vehicles stop
without touching each other. On the right
hand side, you can see the evolution of the vehicles' speed, how the leader
and the follower brake from
the initial speed of 30 m/s until a full stop. Below you can see the relative
speed, which is the difference between the speed of the leader and the follower,
so that you can see the relative speed between these vehicles.
So, this is the solution of the problem.
Here, as further material, we incorporate into the course
and excel spreadsheet where you can play around
with different solutions, with different initial conditions, change
cells in yellow and you can visualize the solution for different case studies.
So, this is everything. Thank you
for watching this video and I hope you enjoy the course.