Work and Energy Legacy Problem #11 Guided Solution

When we think of physics equations, we typically think of them as formulas which serve as a sort of recipe for determining an unknown quantity from a set of given quantities. Yet equations can be much more than that. They can be guides helping us to think about how a variation in one variable would affect another variable. If we take the kinetic energy equation as an example, k e equal one half m b squared. Of course, knowing m and knowing b, we can substitute into the equation and we can solve for k e. That's an equation serving as a recipe for problem solving. But, as I said, an equation can be much more than that. It can guide our thinking about how mass might affect k e, how a variation of the mass would affect the amount of k e an object possesses, or how a variation in the speed would affect the amount of k e that an object possesses. For instance, if you were to double the mass, you would double the amount of kinetic energy. And if you were to double the speed, you would quadruple the amount of kinetic energy. This is because the kinetic energy is directly proportional to mass and also directly proportional to the square of the speed. As such, whatever change is made in the mass, the same exact type of change is made in the kinetic energy. But, whatever change is made in the speed, the square of that change is made in the kinetic energy. We're going to use this principle to approach this question about the kinetic energy possessed by a bicycle and how changes in mass and speed might affect it. So we begin with a bicycle that has a kinetic energy of 124 joules. And the question is, what kinetic energy would it have if it had twice the mass but the speed was unchanged? Since k e is proportional to mass, what we need to do is take the original kinetic energy and double it. 124 multiplied by 2 gives us 248 joules. In part b, they ask, what would be the new kinetic energy if it had the same mass and it was moving with twice the speed? Kinetic energy is directly proportional to the square of the speed. So if you double the speed, you actually cause the kinetic energy to increase by a doubling squared, by a factor of 4. So you take your 124 joules and multiply by 4 to get 496 joules. In part c, we're told that the bicycle has one half the mass and is moving with twice the speed. The fact that the mass has been halved means that the kinetic energy would be halved. But that's not the only change. The speed is doubled. Doubling the speed would quadruple the kinetic energy. So taking into account these two effects, we would have to take 124 and divide by 2 and then multiply by 2 squared. That would give us 248 joules. In part d, we're told that the object has the same mass, but it's moving with one half the speed. The speed has been reduced by a factor of 2, so the kinetic energy gets reduced by a factor of 2 squared. That means we'll have to take the 124 and divide by 2 squared or 4, and that would give us 31.0 joules. Then in part e, again we have two changes being made. One for the mass and one for the speed. The mass is three times as large. If you make the mass three times as large, you'd have to take the 124 and you'd have to multiply by 3. But by also making the speed one half the size, it gets reduced by a factor of 2. Reducing the speed by a factor of 2 would cause the kinetic energy to reduce by a factor of 4. So taking into account these two changes, we would have to multiply by 3 and we would have to divide by 4. And the net effect is that we have 93.0 joules.