9.
Proportionality
Equations are useful things when you want to solve for one of the variables in them, but they also tell you something about the relationship between these variables.Consider the equation
- Because
and are in the numerator on opposite sides, is directly proportional to . That means that if gets bigger, and all the other variables stay the same, then will get bigger too. - The variables
and are inversely proportional, because one is in the numerator on one side, and the other is in the denominator on the other side. That means that if gets bigger, and all the other variables stay the same, then will get smaller. - The variables
and are directly proportional, even though one is in the numerator and one in the denominator, because they are on the same side of the equation. If we solved the equation for , we would see that , and the proportional relationship would be more obvious. - The variables
and are also directly proportional. Furthermore, because is squared, depends more strongly on than it does on or . The variable is doubled if is doubled, but it is quadrupled if is doubled.
- increasing
(i.e. compressing the spring more), - increasing
(i.e. using a stiffer spring), or - decreasing
(i.e. using a less massive ball). However, the radius of the ball doesn’t appear in the equation, so the speed doesn’t depend on that (at least according to this equation).