Kinematics
Why rearrange symbolically
01 / Why rearrange symbolically
Why rearrange symbolically
Why isolate the variable before substituting numbers?
Physics answers are often required in symbolic form, especially in AP Physics. Sometimes a problem provides no numbers at all. You need to be able to solve an equation for the requested variable before a calculator can help.
Substituting numbers too early can hide the algebra, make units harder to track, and create avoidable calculator errors. Write the symbolic rearrangement first, then substitute when the problem actually supplies values.
02 / Reverse PEMDAS
Reverse PEMDAS
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| Look for | Undo it with | Why this comes now |
|---|---|---|
| Addition or subtraction outside the target term | The opposite addition or subtraction on both sides | Separate the target term from other terms first |
| Multiplication or division around the target | The reciprocal operation on both sides | Remove coefficients and grouped factors after outside sums are handled |
| A power, root, or grouped structure containing the target | The corresponding root, power, or carefully justified structural step | Handle the innermost structure only after outside operations are removed |
Scroll horizontally to view every column.
| Where the target appears | Useful first thought | Important caution |
|---|---|---|
| In a denominator | Multiplying both sides by the target often removes that denominator cleanly | Check for other outside additions or subtractions first |
| In a numerator or ordinary product | Keep the target in place and remove the other factors | Treat a grouped product as one quantity |
| In more than one term | Factor the target out before dividing | Do not divide one occurrence away while leaving another behind |
03 / First examples
First examples
Example 1: Isolate time
Solve v = x/t for t.
Assumptions
- The written divisions require t ≠ 0 and v ≠ 0.
- This is symbolic rearrangement; no numerical values or units are being introduced.
Notice that t is in the denominator
v = x/t
Multiplying both sides by t removes the denominator while preserving equality.
Multiply both sides by t
tv = x
The factor of t cancels the division by t on the right side.
Divide both sides by v
t = x/v
Time is now isolated.
Result: t = x/v
Example 2: Simplify common factors before isolating k
Solve ½mv² = ½kx² for k.
Assumptions
- The board-verified starting equation includes m; the automatic captions dropped it from the spoken result.
- The final division requires x ≠ 0.
Cancel the common factor of one-half
mv² = kx²
The factor ½ multiplies every term on both sides, so dividing both sides by ½ simplifies the equation.
Divide both sides by x²
mv²/x² = k
Keep x² together as one factor.
Write the target first
k = mv²/x²
The variable k is isolated.
Result: k = mv²/x²
Example 3: Cancel, remove a fraction, then undo a square
Solve mgh = ½mv² for v.
Assumptions
- The cancellation of m requires the common mass factor to be nonzero.
- The lesson writes the positive square-root form because the target is speed in this algebra context.
Cancel the common factor m
gh = ½v²
Mass multiplies both complete sides.
Multiply both sides by 2
2gh = v²
Multiplying by 2 removes the factor ½.
Take the square root
v = √(2gh)
The square root undoes the square. The expression under the radical must be in the permitted real-number domain.
Result: v = √(2gh)
Check your understanding
Why is simplifying the common ½ in Example 2 better than immediately dividing by x²?
Show the answer
It reduces the number of factors you must track and makes the target term kx² easier to isolate.
Legal simplification before rearrangement often makes later steps shorter and less error-prone.
04 / Denominators and radicals
Denominators and radicals
Example 4: Move a target out of a denominator
Solve v²/r = gh for r.
Assumptions
- The written denominators require r ≠ 0 and gh ≠ 0 where division occurs.
- The symbols are algebra context only; this page does not introduce their physics models.
Multiply both sides by r
v² = ghr
This removes r from the denominator.
Divide by the complete factor gh
v²/(gh) = r
The entire product gh belongs together in the denominator.
Write the target first
r = v²/(gh)
The radius variable is isolated.
Result: r = v²/(gh)
Example 5: Work from the outside of a radical inward
Solve T = 2π√(L/g) for g.
Assumptions
- The written divisions require g ≠ 0 and T ≠ 0 where division occurs.
- The real-number square root requires L/g to be in the permitted domain.
Remove the outside factor 2π
T/(2π) = √(L/g)
The radical cannot be removed cleanly until the factor multiplying it is gone.
Square both complete sides
[T/(2π)]² = L/g
The square applies to the complete fraction T/(2π), not only to T.
Move g out of the denominator
g[T/(2π)]² = L
Multiply both sides by g.
Divide by the complete squared factor
g = L/[T/(2π)]²
This form is already correctly isolated.
Simplify if requested
g = 4π²L/T²
Dividing by T²/(4π²) is equivalent to multiplying by its reciprocal.
Result: g = L/[T/(2π)]² = 4π²L/T²
05 / Factoring and grouping
Factoring and grouping
Example 6: Factor a target that appears in two terms
Solve + = + for .
Assumptions
- The final division requires + ≠ 0.
- Subscripts distinguish different variables and must remain attached to the correct symbols.
Do not move one term to the other side
+ = +
That would leave the target on both sides and make the isolation harder.
Factor out the repeated target
+ = ( + )
Distributing back through the parentheses reproduces the original right side.
Divide by the complete grouped factor
= ( + )/( + )
Keep the numerator sum and denominator sum grouped.
Result: = ( + )/( + )
Example 7: Preserve the grouped numerator
Solve x = t + ½at² for a.
Assumptions
- The final division requires t ≠ 0.
- All three displayed final forms are algebraically equivalent.
Subtract t from both sides
x − t = ½at²
Remove the term added outside the target term before removing coefficients.
Multiply both sides by 2
2(x − t) = at²
The parentheses preserve the entire left-side difference while it is multiplied by 2.
Divide by t²
a = 2(x − t)/t²
Acceleration is isolated.
Result: a = 2(x − t)/t²
Scroll horizontally to view every column.
| Form | How it follows |
|---|---|
| a = 2(x − t)/t² | Keep the numerator factored |
| a = (2x − 2 t)/t² | Distribute the factor 2 |
| a = 2x/t² − 2/t | Split the fraction across the subtraction and cancel one factor of t in the second term |
06 / Reciprocals and final check
Reciprocals and final check
Example 8: Isolate a variable inside a reciprocal
Solve 1/ + 1/ = 1/ for .
Assumptions
- All written denominator quantities must be nonzero.
- The difference 1/ − 1/ must also be nonzero before taking its reciprocal.
Subtract 1/ from both sides
1/ = 1/ − 1/
Do not remove the reciprocal symbols; 1/ + 1/ is not equal to + .
Take the reciprocal of the complete right side
= 1/(1/ − 1/)
The entire difference belongs inside the denominator.
Recognize an equivalent exponent form
= (1/ − 1/)⁻¹
Raising a nonzero quantity to the −1 power means taking its reciprocal.
Result: = 1/(1/ − 1/) = (1/ − 1/)⁻¹
Check your understanding
Before substituting numbers into a rearranged physics equation, what should you verify?
Show the answer
Verify that the requested variable is alone, every step applied an equivalent operation to both sides, grouped sums and products stayed intact, subscripts remained attached to the correct variables, and no illegal cancellation occurred across addition or subtraction.
You can also substitute simple nonzero test values into the original and rearranged forms. Matching results are a useful check, though they do not replace the algebra.
Continue the sequence
Continue learning
This page follows the Isolating Variables math-review lesson in the Kinematics course sequence.
See the Kinematics course