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 tt seconds, we need to integrate the velocity function v(t)=Ze2tv(t) = Ze^{-2t} over the interval from t=0t = 0 to t=tt = t.

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

D(t)=0tv(t)dt=0tZe2tdtD(t) = \int_0^t v(t) \, dt = \int_0^t Ze^{-2t} \, dt

Now, we solve the integral:

D(t)=Z0te2tdtD(t) = Z \int_0^t e^{-2t} \, dt

To integrate e2te^{-2t}, we use the formula for integrating exponential functions:

eatdt=eata\int e^{at} \, dt = \frac{e^{at}}{a}

In this case, a=2a = -2, so:

D(t)=Z[e2t2]0tD(t) = Z \left[ \frac{e^{-2t}}{-2} \right]_0^t

Now, evaluate the definite integral:

D(t)=Z(e2t2e02)D(t) = Z \left( \frac{e^{-2t}}{-2} - \frac{e^{0}}{-2} \right) D(t)=Z(e2t2+12)D(t) = Z \left( \frac{e^{-2t}}{-2} + \frac{1}{2} \right)

Finally, simplify:

D(t)=Z2(1e2t)D(t) = \frac{Z}{2} \left( 1 - e^{-2t} \right)

So, the total distance traveled by the particle during the first tt seconds is:

D(t)=Z2(1e2t) metersD(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 tt approaches infinity?
  3. What would the result be if the velocity function had a different decay rate, say ekte^{-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 tt, 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