Traffic shock wave. An abrupt slowdown in concentrated traffic can travel as a pulse, termed a shock wave, along the line of cars, either downstream (in the traffic direction) or upstream, or it can be stationary. Figure 2-22
shows a uniformly spaced line of cars moving at speed v = 25.0 m/s toward a uniformly spaced line of slow cars moving at speed vs = 5.20 m/s. Assume that each faster car adds length L = 14.0 m (car length plus buffer zone) to the line of slow cars when it joins the line, and assume it slows abruptly at the last instant. (a) For what separation distance d between the faster cars does the shock wave remain stationary? If the separation is twice that amount, what are the (b) speed and (c) direction (upstream denote 1 and downstream denote 0) of the shock wave?
To answer these questions, let's break down the problem and analyze each part step by step.
(a) For what separation distance d between the faster cars does the shock wave remain stationary?
To determine the separation distance d at which the shock wave remains stationary, we need to consider the rate at which the faster cars join the line of slow cars.
Let's assume that the faster cars join the line one after another, meaning that the length that each faster car adds to the line of slow cars is equal to its separation distance from the previous car.
In this scenario, the time it takes for each faster car to join the line is given by the formula:
time = (length added by each faster car) / (speed of the faster car)
time = L / v
In this case, L refers to the length added by each faster car (14.0 m) and v is the speed of the faster cars (25.0 m/s).
The stationary shock wave occurs when the time it takes for each faster car to join the line is equal to the time it takes for the line of slow cars to travel the same distance.
The time it takes for the line of slow cars to travel the distance is given by:
time = (length of the line of slow cars) / (speed of the line of slow cars)
time = d / vs
To keep the shock wave stationary, these two times should be equal:
L / v = d / vs
Now, plug in the given values for L, v, and vs to find the separation distance d.
L = 14.0 m (length added by each faster car)
v = 25.0 m/s (speed of the faster cars)
vs = 5.20 m/s (speed of the line of slow cars)
d = (L / v) * vs
d = (14.0 m / 25.0 m/s) * 5.20 m/s
Now, calculate d.
(b) If the separation is twice that amount, what is the speed of the shock wave?
If the separation distance is twice the value of "d" calculated in part (a), we can substitute the new distance into the formula to find the speed of the shock wave.
Let's define the new separation distance as 2d.
The speed of the shock wave can be calculated using the formula:
speed of the shock wave = (length added by each faster car) / (time taken by the shock wave to travel that distance)
Using the same values of L and v from earlier, and the new separation distance (2d), we can calculate the speed of the shock wave.
(c) What is the direction of the shock wave?
To determine the direction of the shock wave, we need to look at whether it is moving upstream or downstream. In this case, upstream denotes 1 and downstream denotes 0.
Since the shock wave is moving along the line of cars in this scenario, the direction can be determined based on whether the speed of the shock wave is positive (1) or negative (0).
Now, let's calculate the values of (b) and (c) using the given information and the formulas discussed above.