Math Problem Statement
A particle that moves along a straight line has velocity
ひ(t)=ゼe-2t
meters per second after t seconds. How many meters will it travel during the first t seconds (from time=0 to time=t)?
Solution
To find the total distance traveled by the particle during the first seconds, we need to integrate the velocity function over the interval from to .
The distance is given by:
Now, we solve the integral:
To integrate , we use the formula for integrating exponential functions:
In this case, , so:
Now, evaluate the definite integral:
Finally, simplify:
So, the total distance traveled by the particle during the first seconds is:
Would you like further clarification or details on any step?
Here are five related questions to deepen understanding:
- What is the significance of the negative exponent in the velocity function?
- How does the total distance change as approaches infinity?
- What would the result be if the velocity function had a different decay rate, say ?
- How can we calculate the instantaneous acceleration from this velocity function?
- Can we deduce the initial velocity of the particle from the given velocity function?
Tip: When integrating exponential functions, always pay attention to the coefficient of , as it directly affects the integration process.
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Math Problem Analysis
Mathematical Concepts
Calculus
Definite Integration
Exponential Functions
Formulas
Distance traveled: D(t) = ∫ v(t) dt
Exponential function integration: ∫ e^{at} dt = (1/a)e^{at} + C
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
College-level Calculus
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