Solution: Given $ v(t) = t^2 - 4t + mt $ and $ v(2) = 8 $, substitute $ t = 2 $:

February 22, 2026

Title: How to Solve for $ m $ in the Velocity Function $ v(t) = t^2 - 4t + mt $ Using the Condition $ v(2) = 8 $

When working with mathematical models in physics or engineering, functions like $ v(t) $ represent velocity over time. In this article, we’ll explore how to solve for an unknown parameter — $ m $ — in the velocity function $ v(t) = t^2 - 4t + mt $, using a given condition: $ v(2) = 8 $. Substituting $ t = 2 $ is a key step in determining $ m $, and this technique is fundamental in both algebra and applied mathematics.

What is the Problem?

We are given the velocity function: $$ v(t) = t^2 - 4t + mt $$ and the condition: $$ v(2) = 8 $$ Our goal is to find the value of $ m $ that satisfies this condition.

Step 1: Substitute $ t = 2 $ into the Function

To evaluate $ v(2) $, substitute $ t = 2 $ into the expression for $ v(t) $: $$ v(2) = (2)^2 - 4(2) + m(2) $$ Simplify each term: $$ v(2) = 4 - 8 + 2m $$ $$ v(2) = -4 + 2m $$

Step 2: Apply the Given Condition

We know $ v(2) = 8 $. So set the expression equal to 8: $$ -4 + 2m = 8 $$

Step 3: Solve for $ m $

Add 4 to both sides: $$ 2m = 12 $$ Now divide both sides by 2: $$ m = 6 $$

Why This Matters: Application in Real Problems

This method of substituting a known input to solve for a parameter is widely used across disciplines. For example:

In physics, when modeling motion, constants like $ m $ may represent mass or resistance factors.
In economics or optimization, parameters often encode real-world constraints. Solving $ v(2) = 8 $ confirms that $ m = 6 $ ensures the model matches observed data at time $ t = 2 $, validating the equation’s accuracy.

Final Answer

By substituting $ t = 2 $ into $ v(t) = t^2 - 4t + mt $ and applying $ v(2) = 8 $, we determined: $$ m = 6 $$ Thus, the fully defined velocity function is: $$ v(t) = t^2 - 4t + 6t = t^2 + 2t $$ and the condition is satisfied.

Summary Checklist

✔ Substitute $ t = 2 $ into $ v(t) = t^2 - 4t + mt $ ✔ Simplify to $ -4 + 2m $ ✔ Set equal to 8 and solve: $ 2m = 12 $ → $ m = 6 $ ✔ Confirm model accuracy with real-world or analytical context

Learn more about function evaluation and parameter solving in algebraic modeling to strengthen your problem-solving toolkit.

Keywords: velocity function, solve for m, substitute t = 2, algebra problem, mathematical modeling, parameter solving, calculus fundamentals, function evaluation, problem-solving, parameter determination, t² function, m value, v(2) = 8, physics equations, applied mathematics.