To verify the dimensional correctness of the given expression $v_f = v_i + a t$, we analyze the dimensions of each term:

1. Left-hand side (LHS):
   $v_f$ is the final velocity, which has the dimension of velocity:
   [ [v_f] = [\text{L}][\text{T}]^{-1} ]

2. Right-hand side (RHS):

   • $v_i$, the initial velocity, also has the dimension of velocity:
     [ [v_i] = [\text{L}][\text{T}]^{-1} ]
   • $a t$:
     • $a$ is acceleration, which is the rate of change of velocity with time. Hence:
       [ [a] = \frac{[\text{L}][\text{T}]^{-1}}{[\text{T}]} = [\text{L}][\text{T}]^{-2} ]
     • $t$ is time:
       [ [t] = [\text{T}] ]
     • The product $a t$:
       [ [a t] = [\text{L}][\text{T}]^{-2} \cdot [\text{T}] = [\text{L}][\text{T}]^{-1} ]

3. Check dimensional consistency:
   On the RHS, $[v_i]$ and $[a t]$ both have dimensions $[\text{L}][\text{T}]^{-1}$, which matches the LHS.
   Therefore, the equation is dimensionally consistent.

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Related Questions

1. What does it mean for an equation to be dimensionally correct?
2. How can we verify dimensions for other physical quantities, like force or energy?
3. What are the limitations of dimensional analysis?
4. Can a dimensionally correct equation still be physically incorrect? Why?
5. How do the SI base units relate to velocity, acceleration, and time?

Tip:

Dimensional analysis is a powerful tool to check the consistency of equations, but it does not confirm numerical correctness or physical meaning. Always verify with actual units and context!