Find The Distance Traveled In The First Four Seconds For A Particle Whose Velocity Is Given By $v(t) = 2 E^{-t} - \sin T$, Where $t$ Is Measured In Seconds And $v$ Is Measured In $m/s$.

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Introduction

In physics and mathematics, the distance traveled by an object can be calculated using its velocity function. The velocity function gives the rate of change of the object's position with respect to time. In this article, we will find the distance traveled by a particle in the first four seconds, given its velocity function v(t)=2e−t−sin⁡tv(t) = 2 e^{-t} - \sin t, where tt is measured in seconds and vv is measured in m/sm/s.

Understanding the Velocity Function

The velocity function v(t)=2e−t−sin⁡tv(t) = 2 e^{-t} - \sin t represents the rate of change of the particle's position with respect to time. To find the distance traveled, we need to integrate the velocity function with respect to time. However, before we proceed with the integration, let's analyze the given velocity function.

The velocity function consists of two terms: 2e−t2 e^{-t} and −sin⁡t- \sin t. The first term represents an exponential decay, while the second term represents a sinusoidal oscillation. The exponential decay term dominates the velocity function as time increases, causing the particle's velocity to decrease over time.

Integrating the Velocity Function

To find the distance traveled, we need to integrate the velocity function with respect to time. The distance traveled is given by the integral of the velocity function:

s(t)=∫v(t)dt=∫(2e−t−sin⁡t)dts(t) = \int v(t) dt = \int (2 e^{-t} - \sin t) dt

To evaluate this integral, we can use the following properties of integrals:

  • The integral of an exponential function is given by ∫eaxdx=1aeax+C\int e^{ax} dx = \frac{1}{a} e^{ax} + C.
  • The integral of a sine function is given by ∫sin⁡xdx=−cos⁡x+C\int \sin x dx = -\cos x + C.

Using these properties, we can evaluate the integral:

s(t)=∫(2e−t−sin⁡t)dt=−2e−t+cos⁡t+Cs(t) = \int (2 e^{-t} - \sin t) dt = -2 e^{-t} + \cos t + C

where CC is the constant of integration.

Finding the Distance Traveled in the First Four Seconds

To find the distance traveled in the first four seconds, we need to evaluate the integral at t=4t = 4 and subtract the value of the integral at t=0t = 0.

s(4)−s(0)=(−2e−4+cos⁡4)−(−2+1)s(4) - s(0) = (-2 e^{-4} + \cos 4) - (-2 + 1)

Using a calculator or computer software, we can evaluate the expression:

s(4)−s(0)=−2e−4+cos⁡4+1≈1.0003s(4) - s(0) = -2 e^{-4} + \cos 4 + 1 \approx 1.0003

Therefore, the distance traveled by the particle in the first four seconds is approximately 1 meter.

Conclusion

In this article, we found the distance traveled by a particle in the first four seconds, given its velocity function v(t)=2e−t−sin⁡tv(t) = 2 e^{-t} - \sin t. We integrated the velocity function with respect to time to find the distance traveled, and then evaluated the integral at t=4t = 4 and subtracted the value of the integral at t=0t = 0 to find the distance traveled.

The distance traveled by the particle in the first four seconds is approximately 1 meter. This result demonstrates the importance of integrating velocity functions to find distance traveled in physics and mathematics.

Future Work

In future work, we can explore other applications of integrating velocity functions to find distance traveled. For example, we can consider a particle moving in a circular path and find the distance traveled by integrating its velocity function.

References

  • [1] Calculus by Michael Spivak. Publish or Perish, Inc., 2008.
  • [2] Physics for Scientists and Engineers by Paul A. Tipler and Gene Mosca. W.H. Freeman and Company, 2008.

Glossary

  • Velocity function: A function that gives the rate of change of an object's position with respect to time.
  • Distance traveled: The total distance an object has moved from its initial position to its final position.
  • Integration: The process of finding the antiderivative of a function.
  • Exponential decay: A type of decay where the rate of change of a quantity decreases exponentially over time.
  • Sinusoidal oscillation: A type of oscillation where the rate of change of a quantity oscillates sinusoidally over time.

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Q: What is the velocity function given in the problem?

A: The velocity function given in the problem is v(t)=2e−t−sin⁡tv(t) = 2 e^{-t} - \sin t, where tt is measured in seconds and vv is measured in m/sm/s.

Q: What is the distance traveled by the particle in the first four seconds?

A: The distance traveled by the particle in the first four seconds is approximately 1 meter.

Q: How was the distance traveled calculated?

A: The distance traveled was calculated by integrating the velocity function with respect to time. The integral of the velocity function is given by:

s(t)=∫v(t)dt=∫(2e−t−sin⁡t)dt=−2e−t+cos⁡t+Cs(t) = \int v(t) dt = \int (2 e^{-t} - \sin t) dt = -2 e^{-t} + \cos t + C

where CC is the constant of integration.

Q: What is the significance of the constant of integration?

A: The constant of integration, CC, represents the initial position of the particle. When evaluating the distance traveled, we subtract the value of the integral at t=0t = 0 to find the distance traveled from the initial position.

Q: Can the distance traveled be calculated for other time intervals?

A: Yes, the distance traveled can be calculated for other time intervals by evaluating the integral at the desired time and subtracting the value of the integral at the initial time.

Q: What are some real-world applications of finding distance traveled?

A: Finding distance traveled has many real-world applications in physics, engineering, and other fields. For example, it can be used to calculate the distance traveled by a car, a plane, or a spacecraft over a given time period.

Q: Can the velocity function be modified to represent different types of motion?

A: Yes, the velocity function can be modified to represent different types of motion. For example, a sinusoidal velocity function can be used to represent a particle moving in a circular path.

Q: What are some common types of velocity functions?

A: Some common types of velocity functions include:

  • Exponential decay: v(t)=Ae−ktv(t) = Ae^{-kt}
  • Sinusoidal oscillation: v(t)=Asin⁡(ωt+ϕ)v(t) = A \sin(\omega t + \phi)
  • Polynomial: v(t)=at2+bt+cv(t) = at^2 + bt + c

Q: How can the velocity function be used to model real-world phenomena?

A: The velocity function can be used to model real-world phenomena such as the motion of a particle, the flow of a fluid, or the growth of a population.

Q: What are some common applications of velocity functions in physics and engineering?

A: Some common applications of velocity functions in physics and engineering include:

  • Calculating the distance traveled by a particle
  • Modeling the motion of a projectile
  • Calculating the force required to accelerate an object
  • Modeling the flow of a fluid

Q: Can the velocity function be used to model other types of motion?

A: Yes, the velocity function can be used to model other types of motion such as rotational motion, oscillatory motion, or periodic motion.

Q: What are some common types of motion that can be modeled using velocity functions?

A: Some common types of motion that can be modeled using velocity functions include:

  • Rotational motion: v(t)=ωrv(t) = \omega r
  • Oscillatory motion: v(t)=Asin⁡(ωt+ϕ)v(t) = A \sin(\omega t + \phi)
  • Periodic motion: v(t)=Asin⁡(ωt+ϕ)v(t) = A \sin(\omega t + \phi)

Q: How can the velocity function be used to model real-world phenomena in other fields?

A: The velocity function can be used to model real-world phenomena in other fields such as economics, biology, or sociology.

Q: What are some common applications of velocity functions in other fields?

A: Some common applications of velocity functions in other fields include:

  • Modeling population growth
  • Modeling economic systems
  • Modeling biological systems
  • Modeling social systems