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 , the position function, given and , we integrate the velocity function with respect to time and then apply the initial condition to determine the constant of integration.
Step 1: Integrate the velocity function
The integral of is:
Step 2: Apply the initial condition
The problem states that . Substitute and into the equation:
Step 3: Write the final position function
Final Answer:
Do you want a detailed explanation of any step? Here are some related questions:
- What is the connection between position, velocity, and acceleration functions?
- How do we handle initial conditions in integration problems?
- What if the initial condition were different—how would that affect the result?
- Can you explain how to check if is correct by differentiating it?
- 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
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