Math Problem Statement

Find​ s(t), where​ s(t) represents the position function and​ v(t) represents the velocity function.

​v(t)equals

9 t squared

​,

​s(0)equals

3

Solution

To find s(t)s(t), the position function, given v(t)=9t2v(t) = 9t^2 and s(0)=3s(0) = 3, we integrate the velocity function v(t)v(t) with respect to time tt and then apply the initial condition to determine the constant of integration.

Step 1: Integrate the velocity function

s(t)=v(t)dt=9t2dts(t) = \int v(t) \, dt = \int 9t^2 \, dt The integral of 9t29t^2 is: s(t)=9t33+C=3t3+Cs(t) = \frac{9t^3}{3} + C = 3t^3 + C

Step 2: Apply the initial condition

The problem states that s(0)=3s(0) = 3. Substitute t=0t = 0 and s(0)=3s(0) = 3 into the equation: s(0)=3(0)3+C=3s(0) = 3(0)^3 + C = 3 C=3C = 3

Step 3: Write the final position function

s(t)=3t3+3s(t) = 3t^3 + 3

Final Answer:

s(t)=3t3+3s(t) = 3t^3 + 3

Do you want a detailed explanation of any step? Here are some related questions:

  1. What is the connection between position, velocity, and acceleration functions?
  2. How do we handle initial conditions in integration problems?
  3. What if the initial condition s(0)s(0) were different—how would that affect the result?
  4. Can you explain how to check if s(t)s(t) is correct by differentiating it?
  5. How does the constant of integration affect real-world problems in physics?

Tip: Always verify your integration by differentiating the resulting function to ensure it matches the original derivative!

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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Initial Conditions in Differential Equations

Formulas

Integral of a power function ∫t^n dt = (t^(n+1))/(n+1) + C

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12 or Introductory College Calculus