To show that the equation v = v₀ + at is dimensionally consistent, we need to examine the dimensions of each term in the equation and ensure that they are compatible.
Let's break down the dimensions of each term:
- v: Final velocity. Its dimensions are [L][T]⁻¹ (length per unit time), representing velocity.
- v₀: Initial velocity. Its dimensions are [L][T]⁻¹, similar to the final velocity.
- a: Acceleration. Its dimensions are [L][T]⁻² (length per unit time squared), representing the rate of change of velocity.
- t: Time. Its dimensions are [T] (unit of time).
Now, let's analyze the dimensions of each term in the equation:
- v₀ + at:
- The dimensions of v₀ are [L][T]⁻¹.
- The dimensions of a are [L][T]⁻².
- The dimensions of t are [T].
- Adding the terms gives us [L][T]⁻¹ + [L][T]⁻² × [T].
- Simplifying, we have [L][T]⁻¹ + [L][T]⁻¹, which is consistent with the dimensions of velocity, [L][T]⁻¹.
Thus, the dimensions on both sides of the equation are consistent, which means that the equation v = v₀ + at is dimensionally consistent.