a. Find an expression for the minimum stopping distance dₛₜₒₚ of ...

Question

a. Find an expression for the minimum stopping distance dₛₜₒₚ of a car traveling at speed v₀ if the driver's reaction time is Tᵣₑₐ꜀ₜ and the magnitude of the acceleration during maximum braking is a constant a₆ᵣₐₖₑ.

Accepted Answer

Hey, everyone in this problem, we have a trial driving a remote control car at a steady velocity of 3m/s along a straight line. The maximum deceleration of the car is about one m per second. Squared were asked to calculate the shortest possible stopping distance. If the remote control response time is 0.2 seconds, the answer choices were given are a 2.3 m. B 2.9 m C 4. m and D 5.1 m. So this is a Kinnah Matic take problem. Okay. This is a motion problem. We're given information about speed, acceleration and time and we're looking for information about distance. So let's go ahead and write out the variables were given, see what we have a remote control car is driving at a particular velocity V not is equal to three m/s. Hey, we want to find the shortest possible stopping distance which means that we want our final velocity to be 0m/s. Okay, we want that car to come to a stop, which means it's not moving no speed or velocity. The deceleration we're told is about one m per second squared. So when we write this as an acceleration, okay, it's going to be negative because we're decelerating and we have negative one m per second squared. We aren't given information about the time, we're giving some information about the response time, but that's not the total stopping time. And we want to find this distance D alright. So we have three known variables here. One thing we want to find so we can use our kid a Matic or new AM equations with those three known variables to find D we're going to choose the equation that doesn't include the time T and that's gonna be the following V F squared is equal to V naught squared plus two A times the distance D our final speed is zero. So we get zero is equal to three m per second, all squared Plus two multiplied by negative one meters per second squared, multiplied by the distance D simplifying on the right hand side, nine m squared per second squared minus two meters per second squared times the distance D we want to isolate D. So we can move this entire term to the left hand side. We have two m per second squared times the distance D is equal to nine m squared per second squared. We could divide by two m per second squared and we get that the distance D is equal to 4. m. All right, we have to be careful here, we found a distance. The questions asking us for a distance, it can be really easy to stop and say, okay, 4.5 m is the correct answer, right? Remember that this remote control has a response time of .2 seconds. So the distance we found here is a distance from the time the car starts, starts decelerating to the time it stops. But it doesn't take into account the distance occurred, travels during that response time. Okay. So what we need to figure out is the response time, not the response time, but the distance that we traveled during that response time. Okay. So let's do a second set of variables to take this into consideration. Now, before it starts decelerating, this car has a constant speed. So we only have three variables to worry about velocity distance and time. And I'm going to have a subscript art to indicate that this is during the response time. Our car is traveling 3m per second just like it was at the beginning of the deceleration. We're looking for this distance D R And we know that our response time last 0.2 seconds. And when we have constant speed, the equation governing this is going to be that the velocity is equal to the distance over the time. So we have V R is equal to D R divided by T R. And so our distance D R is going to be VR multiplied by T R That's gonna be three m/s, Multiplied by 0.2 seconds. And we get a distance travel during the response time Of 0.6 m. Alright, so we found two distances, we found the distance traveled once the car is decelerating and we found the distance traveled during that response time. And so the total distance traveled that we're looking for is going to be the sum of the two. So our total distance is going to be equal to D plus D R. This is 4.5 m lost 0. m, which gives us a total distance traveled of 5.1 m. Alright, let's go back up and look at our answered traces now that we've answered the question and we found that the shortest possible stopping distance For this remote control car is 5.1 m which corresponds with answer choice D Thanks everyone for watching. I hope this video helped see you in the next one.

a. Find an expression for the minimum stopping distance dₛₜₒₚ of a car traveling at speed v₀ if the driver's reaction time is Tᵣₑₐ꜀ₜ and the magnitude of the acceleration during maximum braking is a constant a₆ᵣₐₖₑ.

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