A robot used in a pharmacy picks up a medicine bottle at t = 0...

Question

A robot used in a pharmacy picks up a medicine bottle at t = 0. It accelerates at 0.20 m/s² for 4.5 s, then travels without acceleration for 68 s and finally decelerates at ―0.40 m/s² for 2.5 s to reach the counter where the pharmacist will take the medicine from the robot. From how far away did the robot fetch the medicine?

Accepted Answer

Hey, everyone in this problem, a car begins its journey from rest by accelerating at a rate of 0.15 m per second squared for six seconds. Afterwards, it maintains a constant speed for 80 seconds. And finally, it decelerates at a rate of negative 0.3 m per second squared for three seconds to come to a stop. We're asked to calculate the distance from stopping location to where the car initially started its journey. We're given four answer choices all in meters. Option A 7.6 multiplied by 10 to the exponent one. Option B 2.7 multiplied by 10 to the exponent. One. Option C 1.3 multiplied by 10 to the exponent two and option D 7.2 multiplied by 10 to the exponent two. All right. So what we're told here is we have three kind of stages and we have this first acceleration stage, we have a constant speed stage and then we have a deceleration stage. OK? So let's write these out. So for stage one, OK, we're told that this car starts from rest. OK? So the initial speed V knot and this is gonna be for stage one. So V knot one is 0 m per second. We don't know the final speed. We don't know how far the car travels. We know that the acceleration is 0.15 m per second squared. And we know that the car travels at this acceleration for six seconds. All right. Ok. So that's our information that we were given now for stage two. What do we have? Well, we're told that it has a constant speed for 80 seconds. Ok. So V two, it's just going to be a constant speed. There's no V not two and VF two, it's just the same speed which we'll call V two. Now, we don't know this speed. They, we aren't told this speed. Ok? But we know that it's gonna be equal to the final speed from stage one. Ok. That car is going to accelerate for six seconds up to some speed and then it's gonna maintain that speed for 80 seconds. Hm. All right. So again, we don't know that speed, but we know that it's that final speed of stage one. We don't know how far the car travels throughout this stage and we know that this stage lasts for 80 seconds. Ok. So T two is 80 seconds and let's go back. We missed the subscript one on T one. All right. Ok. Now, moving to stage three. Ok. We have all of our information about stage two. Now, for stage three. We don't know the initial speed in stage three, but we do know it's gonna be equal to the speed at stage two. Ok? Stage two is a constant speed, then we start to decelerate. So the stage or the speed that we start stage three with is just the speed we were going in stage two, which we also know is this final speed from stage one? Ok. So the feeds throughout these three stages are related to each other. We want this car to come to rest. And so the final speed in stage three is going to be 0 m per second. We don't know how far the car travels. We have an acceleration, a three of negative 0.3 m per second squared and we know that this lasts 437. All right. So what we're really trying to find is the distance from the stopping location. Ok? So we're trying to calculate this delta X value this total distance and that's just gonna be the sum of the three distances for each stage. Ok? So delta X one plus delta X two plus delta X three. Ok. So what we need to do is calculate these individual delta X values for each stage and then add them all together. Now, for stage one, we can go ahead and find delta X one right off the bat. Ok? We have three known values that's gonna allow us to calculate that fourth. So let's do that right off the bat. So we are going to find delta X one. Now we're gonna choose an equation that has the three known values V, not A and T as well as the unknown that we want. Delta X. So we get that delta X one is equal to V, not one multiplied by T one plus one half multiplied by A one multiplied by T one squared substituting in our values. Delta X one is equal to, well, V knot is just zero. So the first term goes to zero, we get one half multiplied by 0.15 m per second squared, multiplied by six seconds, all squared. And this leaves us with a delta X value. OK? One value. Story of 2.7 m. All right. So we have this delta X one value and we are going to want that to calculate our total delta X value. So we're gonna come back to that value. Now, if we want to move to stage two, we can't calculate delta X two until we know V two. Now we know that V two is equal to VF one. OK? That final speed from stage one. So what we can do is use the information about stage one to calculate that final speed and then use that for our other two values. Now looking again at stage one, we have V not A and T this time we wanna calculate VF OK. So we're gonna choose the kinematic equation with those four variables. OK. So finding VF one. Mhm The equation is gonna be VF one is equal to V not one plus a one multiplied by T one substituting in our values. VF one is equal to while V not one is zero. So we just get a 10.15 m per second squared multiplied by T 16 seconds. VF one is going to be equal to 0.9 m per second. OK. All right. So now that we know VF one, can we know the final speed in stage one? That's our speed in stage two, once we stop accelerating. OK. So we two, we now know it's gonna be 0.9 m per second. And similarly, this is gonna be the initial speed for stage three. Once we're done going at that constant speed, that's gonna be our initial speed as we start to decelerate. OK. So we found by finding that one value, we've actually found three values a speed for each stage. All right. Getting back to what we are doing, we are trying to find this total distance. Delta X, we found delta X one. Now, we can find delta X two since we know V two. OK. So we're switching over to stage two and we are gonna find delta X two now because we're moving at a constant speed in stage two, the equation we have to use is just V two is equal to delta X two divided by T two. And velocity is distance over time. This tells us that delta X two, that distance we're looking for is gonna be V two multiplied by T two. This is gonna be equal to 0.9 m per second. That speed we just found multiplied by the time we spend in this stage, which is 80 seconds. And so we get that delta X two is going to be equal to 72 m. All right. So we found two of our three distances. We have one left to find. Let's take a look at the information we have for stage three. OK? We have VNV FA T we have all of the variables except the one we're looking for. So we're gonna choose an equation that will give us that value we're looking for. We wanna have delta X in that equation. So we can solve for. And the equation we're gonna choose is delta X three is equal to V, not three multiplied by T three plus one half, multiplied by a three multiplied by T three squared. Substituting in our values, we have X. Delta X three is equal to 0.9 m per second, multiplied by three seconds plus one half multiplied by negative 0.3 m per second squared, multiplied by three seconds, all squared. And when we work this out, we get delta X three is equal to 1.3 5 m. All right. So we found the distance traveled in each of these three stages. Now we wanna calculate that total distance that we've gone from the beginning of our journey to the end. So the last thing we have to do is add these together. We get that delta X is going to be 2.7 m. Pull off, 72 m, pull off 1.35 m, ok? So this is our stage one, stage two and stage three values added together. And that gives us a final delta X value of 76.05 m. Ok. So that's the total distance we traveled from the time our car started until it came to a stop at the end of the journey. If we compare this to our answer choices and we round to two significant digits, we can see that this answer corresponds with option A 7.6 multiplied by 10 to the exponent 1 m. Thanks everyone for watching. I hope this video helped you in the next one.

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