A Porsche challenges a Honda to a 400 m race. Because the Porsche's acceleration of 3.5 m/s^2 is larger than the Honda's 3.0 m/s^2, the Honda gets a 1.0 s head start. Who wins? By how many seconds?

Video transcript

Hi, everyone in this practice problem. We are being asked to determine who will reach the next traffic light first between the red bike and the black bike. We will have both bikes racing meeting first at the first traffic signal and racing to the next traffic signal which is 600 m away. The red bike is going to start ahead of the black bike by two seconds. And the acceleration of the red bike is 4.5 m per second squared, which is smaller than the acceleration of the black bike, which is 5.5 m per second squared, we're being asked who will reach the next traffic light first and the options are a, both bikes will reach at the same time. B the right bike will reach first C, the black bike will reach first and D we will not be able to determine between both bikes. So in order for us to solve for this problem, we are going to model both bikes as particles and I'm going to start us off with listing a list of known quantities for each bike. So for the red bike, X as knot is going to equal to 0 m. Uh And then V snot is going to equals to also zero m per second because it is starting from rest. And T S Knot is then also going to be zero seconds. Essentially, we want to model that the red bike starting ahead of the black bike by two seconds by saying that the black bike is going to be two seconds late. So X K Knot is also going to be zero m because they are both at the same point or the same traffic light. P K Knot is also going to be zero m per second. And T K Knot is going to be minus two seconds to indicate that it is two seconds late. Um Both X uh both bikes will actually be going 600 m away. So X one is going to be 600 m And XK 1 is also going to be 600 m. The acceleration of the red bike is going to be a S which will equals to 4.5 m per second squared. And the acceleration of the black bike is going to be A K which is going to equals to 5.5 m per second squared. So we will use the position equation to determine the time required by the bike to reach the next signal. So we're going to start with um looking for the time that it takes for the red bike to reach the next signal. So for the red bike, I'm gonna use red to indicate that it is for the red bike or red bike, we want to use the equation of X R or X S one equals X S zero plus V S zero T S one minus T S zero plus half A s multiplied by T S one minus T S zero squared. This will be the equation that we're gonna use and we can essentially start by inputting all of our values except the DS one because that is the value that we are looking for. So the X S one is going to be 600 m equal that with X S zero, which is zero m. I'm just gonna cross it out. P as zero is also zero m. So I'm just gonna neglect this part as well. So both of the uh both the second and the first terms are going to equal to zero. So 600 m is essentially just going to equals to half A S which is 4.5 m per second squared BS one, which is what we are interested in and minus the S zero, which is zero m per second. So this will give us a value of 600 m multiplied by two divided by 4.5 m per second squared, which will equals to T S one squared. So T S one squared is essentially going to be all this value which will actually equals to 266.67 S S squared or second squared. Therefore, the S one is going to be the square root of 266 0.67 S squared, which will equals to 16. seconds just like. So now that we found the time that it takes for a red bike to move to the next traffic light, we are going to tackle the black bite black bike. So for the black bike, we're going to essentially use the same equation. So X K one equals X K zero Plus VK 0 T K, one minus T K zero plus half A K TK 1 - TK 0 Squared. So we know that X K X K zero is equals to zero and V K zero is also equals to zero. So the first two terms can also be neglected. X K one is going to also equal to 600 m and X K 600 m will equals to half A K A K is going to be 5.5 m per second squared. And T K one is going to be the thing that we are interested at and T K zero is going to be minus two seconds squared just like. So, so we will have to actually keep the TK 1 -20. Uh I got a few seconds squared and solve for the constant that would that, that term is going to equals two. So this will be equals to 600 m times two multiplied by two, divided by the acceleration which is 5.5 m/s squared. So TK 1 -2 secondsared is going to equals to 218. S or second squared, which will give us a value of T K one minus negative two seconds equals to the square root of 218.18 seconds squared, which will give us T K one minus negative two seconds to actually be equals to 0.77 seconds. So their four T K one is then going to be 14.77 seconds minus two seconds, which will come out to a value of 12. seconds. So the time that it takes for the black bike to reach the next traffic light is 12.77 seconds. And the time that it takes for the red bike to reach the next traffic light is 16.33 secondss. Therefore, the red bike is going to actually be slower or it takes longer for the red bike to reach the next traffic light. So the black bike will actually be reaching or get to the next traffic light first. So the black bike will get to the next traffic light, which will correspond to option c in our uh answer choices and that'll be all for this particular problem. If you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topics and that will be all for this one. Thank you.