A bicycle coasting at 8.0 m/s comes to a 5.0-m-long, 1.0-m-high ramp. What is the bicycle's speed as it leaves the top of the ramp?

Video transcript

Everyone in this problem. A small metallic box is projected up along a smooth inclined plane with a speed of 2m/s. The length of the incline is 20 cm and its height is 5cm were asked to calculate the boxes speed at the summit of the incline. The answer choices were given are a 0.3 m/s. B 1.4 m per second. C 1.7 m per second and D 1.9 m per second. Let's start by just drawing a little diagram of this problem. So we have the ground, this flat horizontal ground with our metallic box on it And it is moving and we have it moving to the right at 2m/s. And we're going to take the right direction to be positive. And then we have the incline, The length of 20 cm in a fight five centimeters. This is what we have initially, That box is gonna move up the incline. And sometime later, we have the floor, the incline again 20 cm long. We have our box at the summit. This is the same incline 20 cm and five cm height and we want to know what is the speed of the box at this point. So we have two things here. We have this motion of this box and we also have kind of this height of the box, the height of the box changes and the speed of the box changes. So let's consider our conservation of mechanical energy. And we know we have conservation of mechanical energy. What does our mechanical energy consist of? Well, we have the kinetic energy K and we have the potential energy U. So initially, we have K naught plus you not. And then we have that, that is equal to the final kinetic energy K F plus the final potential energy U F. What's our kinetic energy? While our kinetic energy is 1/ and V not square. Can you recall the equation for kinetic energy? We have talked about potential energy here. We don't have any springs or anything like that. So the potential energy is just going to be the potential energy due to gravity, the gravitational potential energy. And recall that that's given by M G H in this case, hr and same thing on the right hand side, kinetic energy one half M V F squared and the potential energy gravitational potential energy M G H F. Now, if we consider this ground where we have our box initially to be H equals zero, then the box has no initial height, the initial height is zero and So it has no initial gravitational potential energy. So that term goes to zero, we also noticed that there's an m the mass in each of these terms, the mass of the block stays the same when it moves. And so we can divide out that mass term from both sides, we're left with one half V, not squared is equal to one half the F squared plus G multiplied by H in the final speed V F. That is that speed that we are looking for. Let's go ahead and substitute in the information we know and see where that gets us. So on the left hand side, we have one half multiplied by two m/s squared. And that initial speed when the block is on the ground, that's equal to one half multiplied by V F square. That quantity we're looking for Plus G, you know, is 9.8 meters per second squared multiplied by the final height. And we know that this incline is five centimeters high, Okay. And so that final height is gonna be five cm. Now, we have to be careful here. Our acceleration is meters per second squared. Our speed is in meters per second. So we want this height to also be in meters, we have centimeters. So we're going to multiply one m per centimeters, the unit of centimeter will cancel or divide them. And what we're doing is essentially taking our cm and dividing by 100 in order to get two m. All right, on the left hand side, let's work this out. We have two squared times a half. This is gonna give us two m squared per second squared. That is equal to 1/2 V F squared plus 0. m squared per second squared. We want to solve for V F. So let's start by moving our 0.49 m squared per second squared to the left hand side. We're just gonna rearrange things so that the, the value we're looking for V F is on the. So we have one half, the F squared is equal to 1.51 m squared per second square multiply, both sides by two V F squared is equal to 3.2 meter squared per second squared. And the last step is to take the square root, we take the square root, We get both the positive and negative route and we get 1.7, 378 meters per second. And we're talking about just as this box is making its way to the incline, we know what's moving to the right. So we know we have a positive speed and so we're gonna take that positive route. So we have that V F is equal to positive 1.7378 m per second. And that is that speed we were looking for. Now, we can go up to our answer choices and compare the answer. We found, we see the answer choices have two significant digits. So to two significant digits, we found that the boxes speed at the summit of the incline was 1. m per second, which corresponds with answer choice. C Thanks everyone for watching. I hope this video helped you in the next one.