A steel ball rolls across a 30-cm-wide felt pad, starting from on...

Question

A steel ball rolls across a 30-cm-wide felt pad, starting from one edge. The ball's speed has dropped to half after traveling 20 cm. Will the ball stop on the felt pad or roll off?

Accepted Answer

Hey, everyone in this problem, a marble ball startss rolling across one edge of a rectangular map with an initial speed v knot. The mat is 40 cm in width. After 18 cm, the balls speed decreased to 1/3 of its initial value, determine the distance traveled by the ball and conclude whether the ball will stay or roll off the mat. All right. So we have this marvel it's rolling. We're given some information about its initial speed and its speed after 18cm. And we want to know is it gonna roll off the map? And in other words, how far does it travel before it stops? And is that distance less than the distance of our map? We have four answer choices. Option, a 20.3 cm. The ball will roll off, Option B 22.4 cm. The ball will roll off, option C 20.3 centimeters. The ball will not roll off and option D 22.4 centimeters, the ball will not roll off. Now, we have information about speed distance. So let's go ahead and write our kinematic variables and see where that gets us. Now, we're gonna write two sets of these. OK? We have one set When we travel that 1st 18 cm and then we have another set when we're traveling the entire distance that this ball rolls. So we're gonna say V, not one, the initial speed is equal to V. They were given that V not in the question. Now, the final speed in this case is going to be one third of the initial value. So the final speed or final velocity is one third A V. In this case, delta X one, the displacement or the distance traveled is 18 centimeters, ok. After traveling 18 centimeters, we reach this speed of one third. V. No, and we aren't given any information about the acceleration and we aren't given any information about the time that this takes. Now, if we look at the entire motion of the marble, ok. The marvel is gonna start with the speed of Von, It's going to end when it comes to rest. So it has a speed of 0m/s. And actually, I'm gonna take the unit out there. We're not sure what the units of V not are that we've been given. So we're gonna leave this just as zero without the units for. Now. Now the distance traveled is what we're looking for. How far does this marble travel? And we're calling that Delta X two. Now, we don't have information about the acceleration of the marble, but we do know that it's gonna be the exact same as the acceleration in the first part and the marble is accelerating at some rate. And that's gonna be the same whether we're looking at the first half of the marvel rolling or the entire time, the marvel is rolled and we aren't given any information about the time. So we wanna find delta X two, but we don't have enough known values in this second set of variables In the first set of variables, we have VA V F one and delta X one that we have at least some information about. So let's go ahead and use those to calculate the acceleration A one. And then we can use the acceleration A one in our equation in this second part. OK? Because the acceleration is the same. So we want to choose a kinematic equation that has the four variables V, not one V F one, delta X one and a one. OK? We're choosing the one that doesn't have the time because we aren't given information about that. And that's not what we're looking for. We have V F one squared is equal to V, not one squared plus two A Delta X one substituting in the values for the information, we know we have one third, V, not all squared Is equal to V, not squared plus two A one Multiplied by 18 cm. We have one night V not squared Is equal to V, not squared plus two A one Multiplied by 18 cm. Now we Wanna isolate for a one. OK. We want to solve for a one. So the first thing we have to do is subtract V not squared from both sides. So we have 1/9 V, not squared -3, not squared. And then we need to divide by this two multiplied by 18cm. We're gonna divide By two, multiplied by 18 cm. Simplifying the numerator, this gives us negative 8/9 V not squared Divided by two multiplied by 18 cm. And we can simplify this will give us negative to V not squared divided by 81. OK? We'll leave it as a fraction for now. So we don't have to do any rounding. Now, where does that get us in this problem? Let's go back up. We're trying to find delta X two. We now know this acceleration A one. OK? Which is equal to the acceleration A two at least in terms of va. So let's switch over to the second equation or the second set of variables. See where we can get with that. Now, we're gonna choose the same kinematic equation because we have the same four variables. We're ignoring the time T We have VF two squared is equal to V, not two squared plus two A two, Delta X two. Then again, Delta X two, that's what we're looking for. Now, the final speed is zero. OK? When that ball comes to rest the initial speed V not squared plus two multiplied by the acceleration which we found to be negative two V not squared divided by 81 Multiplied by Delta X two. We can simplify, we can write this as zero is equal to V, not squared minus four V, not squared divided by 81 Multiplied by Delta X two. We want to solve for delta X two. So let's move this entire term with delta X two to the left hand side, we have four V not squared divided by 81 is equal to oops, we forgot our delta X two multiplied by delta X two is equal to V. Not squared no Units here. OK. When we go back to our acceleration, a one, we had two multiplied by 18 cm in the denominator. And so we actually have a unit of 81 cm in the denominator here. OK? All right. Now we have V not squared on both sides. We can divide by V not squared. OK. That term divides out, We're left with four divided by 81 cm multiplied by Delta X two is equal to one. Now, we can solve for Delta X two And it's going to be equal to cm divided by four, Which is equal to 20.2, 5 cm. OK. So we needed to make sure we followed that unit of centimeters all the way through this problem so that we got the correct units in the end. And what's interesting here is we didn't need to know that initial speed. Ok. We didn't need to know what V NAT was. We just needed to continue using V NAT throughout the problem. And eventually we were able to divide that value out. So we found that the marble is gonna travel a distance of 20.25 cm. Remember that the map is 40 cm. So if the marvel is traveling 20.2, 5 cm, it's still going to be on the map, it hasn't traveled further than the distance of the map. And so rounding to three significant digits, the correct answer is answer choice. C the marble will travel 20.3 cm and will not roll, roll off the map. Thanks everyone for watching. I hope this video helped see you in the next one.

Struggling with Physics?

A steel ball rolls across a 30-cm-wide felt pad, starting from one edge. The ball's speed has dropped to half after traveling 20 cm. Will the ball stop on the felt pad or roll off?

Watch next

Vectors, Scalars, & Displacement practice set