Check the dimensional correctness of s = ut + ½at²
Second equation of motion. The distance covered in time t with a starting speed u and a steady acceleration a.
Equation checker
Matching dimensions are necessary, not sufficient. A check cannot see pure numbers such as 1/2 or 2π, so a consistent equation can still be wrong.
s = u t + (1/2) a t2s[L]=ut[L]+1/2 at2[L]Step by step
1. What each symbol stands for
- s displacement [L]
- u velocity [L T-1]
- t time [T]
- a acceleration [L T-2]
2. The dimensions of each term
- s [L] [L]
- ut [L T-1] × [T] [L]
- +1/2 at2 1/2 × [L T-2] × [T]2 [L]
3. Compare
Every term has dimensions [L], so the equation is dimensionally consistent.
What it proves and what it does not
It can show
- Every term that is added or subtracted has the same dimensions, so the sum makes sense.
- Both sides of the equals sign have the same dimensions.
- Angles inside sin, cos and similar functions are pure numbers.
It cannot show
- A pure number such as 1/2 or 2π has no dimensions, so a check cannot see a missing or wrong one.
- Two quantities with the same dimensions, such as work and torque, cannot be told apart.
- A sign error or a wrong power that keeps the dimensions equal goes unnoticed.
Matching dimensions are necessary but not sufficient. Pure numbers such as 1/2 or 2π have no dimensions, so a check cannot see them, and an equation with matching dimensions can still be wrong.
Look up each quantity
s = ut + ½at²: quick answers
Is s = ut + ½at² dimensionally correct?
Yes. Every term has the same dimensions, [L], so s = ut + ½at² is dimensionally consistent. That does not prove the equation itself is correct.
Does a dimension check prove s = ut + ½at² is right?
No. A check can show that an equation is wrong, but it cannot show that it is right. Pure numbers such as 1/2 and 2π have no dimensions, so the check cannot see them.