A car traveling at 30 m/s runs out of gas while traveling up a 10...

Question

A car traveling at 30 m/s runs out of gas while traveling up a 10 degree slope. How far up the hill will it coast before starting to roll back down?

Accepted Answer

Hey, everyone in this problem, a skateboard released with an initial speed of 10 m/s, moves up an inclined plane. The inco plane makes an angle of 15° with the horizontal. We're asked to determine the distance traveled before the skateboard changes its direction of motion. Ok. So we're looking for the distance traveled. And the answer choices we're given are a 5.1 m. B 5.28 m C 19.7 m and D m. So let's go ahead and draw a little diagram to get an idea of what we have. So we have an inclined plain and out this. So it's straight. We're told that the angle between the incline plane and the horizontal is 15°. We have a skateboard. So we'll draw a little skateboard going up our incline. And we're told that the initial speed of the skateboard, which we'll call v knot is 10 m/s. Now, we're gonna take to the right to be our positive direction and we're gonna take the tilted axis can that runs parallel to this plane. Yeah, we're gonna call this our X axis for the purpose of this problem. Ok. So that's gonna be our axis and we wanna find the distance this travels, the skateboard travels before it comes to a stop. So sometime later it's going to be at this maximum height before it changes direction and the speed is gonna be zero m per second there. Ok. The skateboard is gonna come to rest momentarily before it changes direction and comes down. Ok. All right. So we're looking for a final speed of zero and we wanna know the distance traveled between these two points. We're gonna call that delta. Now we have some speeds. We're looking for distance. So let's go ahead and write up the variables we have or our kinematic equations or our U AM equations if that's how you like to call them. So our initial speed VNA is 10 m per second. Our final speed VF is 0m/s. We wanna find delta X. OK. That's that distance that the skateboard has traveled. The acceleration. Now, we're not told the acceleration. Well, we do have gravity acting because the skateboard is on an incline plane, it's reaching up to a height. So we have gravity acting on this skateboard. And if we think about it, I'm gonna draw another diagram. So we have our plane here. We have our skateboard and gravity is going to act straight downwards, which means that the acceleration, OK? We can project this gravity onto that plane the acceleration we're looking for, it's going to be acting to the left. OK. This indicates it's going to be a negative acceleration. And what is the magnitude We know the angle between this plane and the horizontal is 15° That tells us that the angle on the top right. OK. Between the acceleration and gravity is going to be 90° -15°. And so the angle between the gravity and the vertical is going to be 15° as well. All right. So using some triangle math, we can write that the acceleration A is going to be equal to negative sign of 15° Times G. And again, we have that negative because we've chosen up into the right as positive. OK. So this acceleration is pointing to the left. So this is a negative acceleration. And so we have negative sign of 15° and the gravitational acceleration is 9.8 m/s squared. And we aren't given any information about the time and that's not what we want to figure out. So we have three known values V, not V F and A, we're looking for the distance travel delta X. So we can go ahead and choose the kinematic equation that doesn't include the time T and that includes the other four variables plug in our values and solve for delta X. If we do that, we're gonna have the equation V F squared is equal to B not squared. Plus two multiplied by a multiplied by Delta X substituting in our values, we have zero is equal to 10 m per second squared plus two, multiplied by negative sign of degrees Multiplied by 9.8 m/s squared multiplied by delta X. And we can move this entire term, the right-hand term to the left hand side. OK? When we do that, it's a negative. So it's gonna become a positive. We have two times 9.8, which is gonna give us 19.6. And so we have 19.6 m per second squared times sign of 15 degrees multiplied by delta X is equal to m per second squared. It's gonna be 100 m squared per second squared. All right, we want to solve for delta X. So we're gonna divide, Can we get 100 meters squared per second squared divided by 19. m per second squared Multiplied by sine of 15°, which gives us a delta x value of 19.713 m. OK? And that is that distance traveled along the incline for the skateboard that we were looking for. We go back up to our answer choices. We see that they have three significant digits and to three significant digits. We found the distance traveled to be 19.7 m that corresponds with answer choice C Now I wanna make one more note here before we go this problem, we've chosen to solve with kinematic, using our kinematic equations, you can also solve this using the conservation of mechanical energy. So if that's a topic that you've learned already, that would be great practice to go through and solve this problem with conservation of mechanical energy. And you're gonna get that same result. Thanks everyone for watching. I hope this video helped see you in the next one.

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A car traveling at 30 m/s runs out of gas while traveling up a 10 degree slope. How far up the hill will it coast before starting to roll back down?

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