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)?

To find the total distance traveled by the particle during the first \( t \) seconds, we need to integrate the velocity function \( v(t) = Ze^{-2t} \) over the interval from \( t = 0 \) to \( t = t \).

The distance \( D(t) \) is given by:

\[
D(t) = \int_0^t v(t) \, dt = \int_0^t Ze^{-2t} \, dt
\]

Now, we solve the integral:

\[
D(t) = Z \int_0^t e^{-2t} \, dt
\]

To integrate \( e^{-2t} \), we use the formula for integrating exponential functions:

\[
\int e^{at} \, dt = \frac{e^{at}}{a}
\]

In this case, \( a = -2 \), so:

\[
D(t) = Z \left[ \frac{e^{-2t}}{-2} \right]_0^t
\]

Now, evaluate the definite integral:

\[
D(t) = Z \left( \frac{e^{-2t}}{-2} - \frac{e^{0}}{-2} \right)
\]
\[
D(t) = Z \left( \frac{e^{-2t}}{-2} + \frac{1}{2} \right)
\]

Finally, simplify:

\[
D(t) = \frac{Z}{2} \left( 1 - e^{-2t} \right)
\]

So, the total distance traveled by the particle during the first \( t \) seconds is:

\[
D(t) = \frac{Z}{2} \left( 1 - e^{-2t} \right) \text{ meters}
\]

Would you like further clarification or details on any step?

Here are five related questions to deepen understanding:
1. What is the significance of the negative exponent in the velocity function?
2. How does the total distance change as \( t \) approaches infinity?
3. What would the result be if the velocity function had a different decay rate, say \( e^{-kt} \)?
4. How can we calculate the instantaneous acceleration from this velocity function?
5. 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 \( t \), as it directly affects the integration process.

A particle that moves along a straight line has velocity v(t) = Ze^{-2t} meters per second after t seconds. How many meters will it travel during the first t seconds (from time=0 to time=t)?

Discover how to calculate the total distance traveled by a particle with velocity v(t) = Ze^{-2t} over the first t seconds. This solution involves the integration of an exponential function and covers key calculus concepts, including definite integration and the fundamental theorem of calculus.

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