Some Math Review

 

Unit Conversion

Multiplying any number by 1 leaves the number the same. Dividing any number by itself (except 0) and you get 1.

 

12 = 12

 

   12 / 12 = 1

 

3 = 3 x 1 = 3 x (12 / 12) = 36 / 12 = 3

 

36 / 12 is the same thing as 3, even though it looks different. Similarly, we can convert a physical quantity measured in some units to another set of units and still have the same thing. Example: convert 73.4 cm to feet:

 

1.0 inch = 2.54 cm            1.0 in / 2.54 cm = 1

 

1.0 foot = 12 inches            1.0 ft / 12 in = 1

 

73.4 cm = 73.4 cm x       1 in    x      1 ft     = 2.41 ft.

                                    2.54 cm        12 in

 

“Trick”: Always set up ratios so units cancel algebraically.

General Solutions

The simplest mathematical solution to a problem that can describe and predict the behavior of the physical world is the general solution that we are looking for in our problem solving. For example:

 

Suppose you want to find the mass, m, of a projected ball that lands on and compresses a spring (spring constant k) a distance, x, if thrown downward from height, h, at speed, v. From conservation of energy, you come up with:

 

            ½ kx2 = ½ mvo2 + mgh

 

But, in the given problem, you want the mass, m. You must then solve this equation for m.

 

Multiply through by 2:          kx2 = mvo2 + 2mgh

 

Factor out the m:        kx2 = m (vo2 + 2gh)

 

Isolate m:        m = kx2 / (vo2 + 2gh)

 

Not only do you now have a single expression that explicitly solves for m, but also you can easily see how the quantity you solved for (m) depends on other variables.
Simultaneous Equations

There are plenty of physics situations where you have more than one unknown. For two unknowns, you’ll need two equations, for three unknowns, three equations, etc…

 

A simple example:         j + k = 4                      (1)

                                          2jk + k = 10              (2)

 

Rewrite (1) in terms of k:           k = 4 – j

 

And sub into (2):            2j(4 – j) + (4 – j) = 10

 

Continue:                   8j – 2j2 + 4 – j = 10

 

Bring over 10, combine terms, multiply through by –1:

 

                                    2j2 – 7j + 6 = 0

 

A quadratic equation.


Simultaneous Equations continued

 

The general solution to a quadratic equation is (excusing the poor layout of this equation):

 

                              j = (-b + (b2 – 4ac)^ ½) / 2a

 

In this case, a = 2, b = -7, c = 6.

 

The “plus” solution:  

j = (7 + (49 – 48)^ ½) / 4 = 8 / 4 = 2.  Then k = 2.

 

The “minus” solution:

          j = (7 – (49 – 48)^ ½) / 4 = 6 / 4 = 1.5.  Then k = 2.5.

 

The end.

 

OR, instead, sometimes you can factor:

 

      2j2 – 7j + 6 = 0  =>  (2j – 3) (j – 2) = 0.

 

Either (2j – 3) = 0 thus j = 1.5       (and k = 2.5), or

           (j – 2) = 0 thus j = 2      (and k = 2).