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Along a straight road through town, there are three speed-limit signs. They occur in the following order: $55,35,$ and 25 $\mathrm{mi} / \mathrm{h}$ , with the $35-\mathrm{mi} / \mathrm{h}$ sign located midway between the other two. Obeying these speed limits, the smallest possible time $t_{\mathrm{A}}$ that a driver can spend on this part of the road is to travel between the first and second signs at 55 $\mathrm{mi} / \mathrm{h}$ and between the second and third signs at 35 $\mathrm{mi} / \mathrm{h}$ . More realistically, a driver could slow down from 55 to 35 $\mathrm{mi} / \mathrm{h}$ with a constant deceleration and then do a similar thing from 35 to 25 $\mathrm{mi} / \mathrm{h}$ . This alternative requires a time $t_{\mathrm{B}} .$ Find the ratio $t_{\mathrm{B}} / t_{\mathrm{A}}$

Solved $\frac{t_{b}}{t_{a}},$ the result is 1.19

Physics 101 Mechanics

Chapter 2

Kinematics in One Dimension

Motion Along a Straight Line

Cornell University

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all right. In this problem, we have a car that is driving down a road. And on this road, the speed limit goes from 55 miles an hour down to 35 miles an hour and then all the way down to 25 miles an hour. And this problem is asking us to find, uh, how the time traveling down this road compares to a car that is traveling at the maximum speed for both of those two sections, both path number one and path number two compares to the total time it takes a car that is gradually slowing down during the entire process. We're looking for the ratio of how those to compare now in order to do this, where you're going to turn to our kin O Matic equations. Here we have our five variables where we need to know three of them and were to plug in numbers. Now, one of the interesting things here is that we don't actually know enough to use all the Kinnah Matic equations. For example, in this first part where we're looking at it in a case where we're traveling at the full speed for the entire time, We know that there's no acceleration going along this road. We're saying they're staying at a constant velocity for both of these parts and then kind of magically slowing down when they hit the speed limits. Hide. So for Section one, we're traveling at 55 meters per second per section. Two were traveling at 35 meters per second and we don't ever change the speeds outside of that. Now we're looking for the time it takes to go that speed down this road, but we don't know how long the road is. So the interesting thing here is that this problem is all about ratios. So although we don't know the distance, this road is, it also doesn't matter because whatever distance the road is is going to get cancelled out and the ratio of the times will stay the same. So, for ease of problem solving, I'm going to say, Let's say both of these sections of road are 100 meters and really, you could plug in whatever number you want here because the ratio part isn't going to change. So we're looking at this same value, got so they're both 100 meters because we know the signs are evenly spread out. So we need to figure out what our times are going to be. And we have both time one here and time to in my end goal for this first part is I want to know when I add those two together. What is the total time for this path? Time a going to be. So since we're not accelerating, we can actually just use, um, our equation. Um, she looking over the side here. We don't care about the final velocity. Um, so we're gonna be using the last equation of the bottom X equals V, not T plus 1/2 A T squared. And since the acceleration is zero, that last part kinda drops off for us. So we have X equals V, not tea. And since we're solving for tea, I will divide that initial velocity, and we get distance divided by V not will give us our time. So keeping that in mind, let's plug our numbers in here and get our t one and t two's out. T one is equal to the distance of 100 divided by the velocity of 55 t two is equal to the 100 the same distance divided by the velocity of 35. And that gives us the two individual times of 1.82 seconds for that first portion and for the second portion we have 2.86 seconds. When I add those two together to get the time of my first piece, we get 4.68 seconds and that's how long it would take to drive this with absolutely no acceleration. And so second part is looking for what happens if we do that same problem. The same set up the same everything, except this time we're gradually accelerating from our top speed to our bottom speed. So my first section, we're starting at 55 miles or meters per second and we are ending at 35 meters per second and the second part, I am starting at 35 meters per second and ending at 25 meters per second. Now. The other piece is going to stay the same because I said let's assume it's 100 meter long road for the other one that stays the same for this one. Um, we're looking for the times, both the time for the first section and the time for the second section. So this acceleration is not something we're going to work with, which means when I'm picking my equation, I'm looking for the one that does not have an acceleration in it. Which is the third equation there. That X equals 1/2 V plus B, not times t so like it before. I want to actually solve this before we start plugging things in because we're just finding time for both of these. So that'll save us a little bit of time to rearrange it first. So I'm going to divide by the V plus B not and multiply by two. To simplify this to the time is equal to two times the displacement divided by B plus V not. And when I go to plug in my answers here I have t one is equal to 200 divided by 35 plus 55 and tea, too, is 200 divided by 35 plus 25. I put those into a calculator. We have a T one of 2.22 seconds and 82 of 3.33 seconds. So in total the time. It takes me to go down. This second path where I have an acceleration the whole time is 5.55 seconds. Now, the last part of this problem is asking me to put those together to figure out the ratio of time. Be two time, eh? So when I go to my last slide here, I have time be a 5.55 time A of 4.68 Since this problem wants me to do time of be compared to time of a, they can put those together. Toe have 5.55 divided by 4.68 and this ends with a total time of 1.19 seconds.

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