Newton's Laws Legacy Problem #10 Guided Solution

You know, we often think of equations as a sort of recipe which prescribes how we might calculate one mathematical quantity from several other mathematical quantities. For instance, A equal F net over M can be used to calculate an acceleration if we know F net and we know M. But I would insist that equations are much more than recipes for problem solving. They're actually guides to thinking about how an alteration in one of our variables might affect another variable. For instance, the equation A equal F net over M would tell me that the acceleration is directly proportional to the F net and that whatever change is made in the F net, the same change would be made in the acceleration. So a doubling or a tripling or a quadrupling of F net has the same effect upon the acceleration. And similarly, A equal F net over M tells me that acceleration is inversely proportional to mass. Whatever alteration is made in the mass, the inverse alteration is made in the acceleration. So think about this now. If you double the mass, the acceleration is going to decrease by a factor of two. We'd say it is halved. If you triple the mass, the acceleration becomes one third of the original value. If you half the mass, make the mass one half as big, then you're going to make the acceleration two times as big. So we can use equations as guides to thinking about how an alteration in one variable would affect the other variable. And that's exactly what I'm going to do here in this problem. In part A, in the problem, I'm told that a force of F causes a mass of M to accelerate at 48. So 48 centimeters per second per second is my original acceleration. What if I make the force twice as much as in part A? Well, the acceleration would be twice as much. I have to double the 48. In part B, what if I made the mass twice as much? The mass, the denominator of the A left over M equation. Well, if you make the denominator twice as big, you have to make the acceleration one half as small. So you have to take the 48 and multiply by one half, or divide by two. In part C, I'm told that the net force doubles, well, that doubles the numerator. And I'm told that the mass doubles, well, that doubles the denominator. These two changes offset one another such that the acceleration remains the same as it was. And in part D, I'm doubling the force, 2F, and I'm making the mass four times as big. So the numerator gets twice as big, and the denominator four times as big. That's a factor of 2 over 4. My new acceleration is two fourths, or one half, of 48. And finally, in part E, what I know is the numerator of the force is four times as big, and the mass is two times as big. That's a ratio of F to M of 4 over 2. I'm increasing the F-M ratio by 2. I have to increase the acceleration by a factor of 2. That's how you use equations. This guides the thinking.