Here is a typical problem from one-dimensional kinematics: A car accelerates from 20 m/s to 45 m/s over a distance of 82 m. What is its acceleration value?
The equation of choice is:
Note that when no operation symbol appears between two characters in an equation, that implies multiplication. When the numbers are plugged into the equation, we get
We solve for the acceleration by gradually isolating "a" on one side of the = sign. Here are the algebraic steps followed by two students on this problem during a quiz:
| (45 - 20)2 = 2a(82)
252 = a(164) 625 = a(164) 3.8 = a |
2025 = 400 + 2a(82)
(2025 - 82) = (400 - 82) + 2a 1943 = 318 + 2a 1625 = 2a 812.5 = a |
Both are wrong, for different reasons.
The first one is wrong because a squared number can't be involved in another operation prior to either removing the square or completing the square operation. Squaring (multiplication of a number by itself) is a higher-priority operation than addition. In general, multiplication takes priority "over" addition.
The second is wrong because the entire term 2a(82) is added to the 400; the 82 is not separable in this way. Also, we would remove the 82 from the 2a(82) once it is all by itself on one side of the = sign, by dividing all terms by 82 rather than subtracting, since dividing undoes the multiplication operation.
The correct procedure involves starting on the terms farthest away from the one containing the unknown "a"; i.e. working our way toward "a" from the outside.
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| To help you remember the correct technique: think of squared or multiplied numbers as shielded from the operations of addition or subtraction. The "shield" has to be removed before you can add or subtract, or you have to leave the "shield" intact until the term is isolated. | |
Practice with these examples:
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6(16) - 2(b + 5) = 12 solve for b
2(c2 - 32) = 4 solve for c 12 = 6d + ½(5)(32) solve for d |
answers: b = 37 c = square root of 11 d = -1.75