Math Problem Statement
The velocity of a particle moving in a straight line is given by v(t) = 5t + 18t^2. If s(3) = 8, where s(t) is the distance traveled, find s(5)
Solution
To solve this problem, we need to find for the given velocity function , and the initial condition .
Steps:
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Relation Between Velocity and Distance: Velocity is the derivative of the distance function :
Therefore, the distance function is obtained by integrating :
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Integrate the Velocity Function: Given , integrate it: where is the constant of integration. Simplify:
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Apply the Initial Condition : Substitute and into the equation to solve for : Simplify the terms: Combine: Solve for :
Thus, the distance function is:
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Find : Substitute into : Simplify each term: Combine terms:
Final Answer:
Would you like more details or have any questions?
Related Questions:
- How would the solution change if had a different value?
- Can you explain how the integral is related to velocity and displacement?
- What would represent in this context?
- How would the function change if were instead ?
- What if the particle moved in a circular path instead of a straight line?
Tip:
Always check your integration constants carefully when solving problems involving definite or indefinite integrals.
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Math Problem Analysis
Mathematical Concepts
Velocity
Distance
Integral Calculus
Differentiation
Formulas
v(t) = ds(t)/dt
s(t) = ∫v(t) dt
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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