A motorist is driving at 20 m/s when she sees that a traffic light 200 m ahead has just turned red. She knows that this light stays red for 15 s, and she wants to reach the light just as it turns green again. It takes her 1.0 s to step on the brakes and begin slowing. What is her speed as she reaches the light at the instant it turns green?

Video transcript

Hey everyone in this problem. A bus is driven at a speed of 18 m/s. The bus driver notices a competitor van at a bus stop m ahead. The van schedule states it will leave the bus stop after 16 seconds. The bus driver which wishes to reach the bus stop as the van takes off, the driver has a response time of two seconds before applying brakes and decelerating uniformly and were asked to determine the speed of the bus as it arrives at the bus stop while the van is taking off. Our answer choices here are a 2.6 m per second. B 0.4 m per second. C negative 1.8 m per second and D 15.8 m per second. Alright, so we want to figure out the speed of the bus as it arrives at the bus stop. We have to kind of stages to consider here We have stage one okay before the bus applies brakes. Now, in this stage, the bus is not accelerating, it's driven at a constant speed. And so the only three variables we have to consider are the speed the distance and the time. So our speed They were given in the problem is m/s. We don't know how far we travel, but we know that this period lasts two seconds. It takes two seconds of response time before the driver starts to apply the brakes. So this period before they apply the brakes is two seconds. Now, we also have a second stage stage two and let me call this V one D one D two and stage two is going to be when the bus is decelerating. And in the decelerating phase, what do we have? We know that the bus starts decelerating from m/s. We don't know what the final speed will be, but that's what we want to find out. We don't know the acceleration, we know that it's decelerating. We know the acceleration therefore will be negative but we don't know the value of a don't know the distance to, we know how far the bus stop is ahead 180 m. But we also have to account for the distance it travels in stage one. And so the distance we travel in stage two is actually going to be the 180 m minus the distance we travel in stage one. Alright. And then our time T Which is gonna be T2, we know the van will leave after 16 seconds. The bus is going to arrive right when the van is leaving. So the total time is going to be 16 seconds But two of those seconds are going to be spent in stage one before we apply the brakes. So we have to subtract two seconds. So the time we spend actually decelerating is going to be 14 seconds. So if we look at this, we know V not and we know T two, okay, we know some information about D too, but we don't have a numerical value. And so if we're looking just at stage two to try to find the F, we don't have enough information to use. Are you am or Kinnah Matic equations were decelerating uniformly. So we can use those equations, but we don't have three known quantities in order to use them. So what we can do is go back to stage one, we can use the information here to figure out what D one is. And then we have a relationship between D two and D one. If we know D one, then we know the distance we travel D two for stage two. And we can find via. So in stage one, we're just moving at a constant speed. And so we can consider the equation they recall V is equal to distance over time. So we have V one is equal to D one over T one, The speed is 18 m/s. We're looking for the distance and the time is two seconds. So this tells us that the distance traveled in that first stage before the bus driver stops, the brakes is going to be 18 m/s Times two seconds for a total distance of 36 m. Alright, so then our distance D2 Okay that we travel, that we decelerate Is going to be equal to that total m distance from the bus to the bus stop minus the 36 m we traveled before decelerating. Which is gonna give us 144 m. All right. Now we have Vienna, we have D two and we have T two. Now we have three notes. We can find that fourth quantity we're going to choose are you am equation that doesn't include acceleration. We don't know the value of the acceleration and that's not the quantity we're looking for. So we're gonna go ahead and choose the equation without acceleration. And that is going to be the following equation. Delta X is equal to 1/2 times vino VF times T Delta X. That's the same as the distance traveled, that change in position, that distance that we travel. So 144 m is equal to one half our initial speed of 18 m per second plus the final speed. We're looking for times the time 14 seconds, we have one half times 14 seconds gives us seven seconds. So 144 m, we can divide by that seven seconds. Is going to equal 18 m per second plus the final speed V F isolating the F we can move this 18 m per second to the left hand side. We have that V F is equal to 144 m divided by seven seconds minus 18 m per second. If we work this out, we get approximately 2.57 m per second for our final speed. Alright and that's the final speed of the bus we were looking for when it arrives at the bus stop. Now, I want to make a quick comment here. Most professors in most textbooks will use this Kinnah Matic equation, this fifth equation without the acceleration. There are some cases where your professor won't want you to use this equation or where the textbook won't include this equation. If that's the case for you, what you can do is use the known values V not D two and T to to calculate the acceleration first and then use that to calculate V F. Okay. So if that's the case that your professor does not want you using this equation, which is rare, but it can happen then there is still a way to solve this problem. It's not dependent on that equation. Alright, let's go back up to our answer choices. And we see that we have found the speed of the bus to be approximately 2.6 m/s, which is answer choice. Thanks everyone. For watching. I hope this video helped see you in the next one.