A Particle Travels Along The \[$ X \$\]-axis Such That Its Velocity Is Given By $ V(t) = T^{0.8} + 5 \sin (3t - 1) $. If The Position Of The Particle Is \[$ X = -5 \$\] When \[$ T = 1.5 \$\], What Is The Position Of The

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Introduction

In physics, the motion of an object can be described using various mathematical equations. One of the fundamental equations is the relationship between velocity and position. In this article, we will explore how to calculate the position of a particle given its velocity function. We will use the velocity function $ v(t) = t^{0.8} + 5 \sin (3t - 1) $ and the initial position $ x = -5 $ at time $ t = 1.5 $ to find the position of the particle at any time $ t $.

Understanding Velocity and Position

Velocity is the rate of change of an object's position with respect to time. It is a vector quantity, which means it has both magnitude (speed) and direction. The position of an object, on the other hand, is a scalar quantity that describes its location in space. In this article, we will focus on the position of a particle along the x-axis.

The Relationship Between Velocity and Position

The relationship between velocity and position is given by the equation:

x(t)=v(t)dt x(t) = \int v(t) dt

where $ x(t) $ is the position of the particle at time $ t $, and $ v(t) $ is the velocity function.

Calculating the Position of the Particle

To calculate the position of the particle, we need to integrate the velocity function $ v(t) = t^{0.8} + 5 \sin (3t - 1) $ with respect to time $ t $. We can do this using the following steps:

  1. Integrate the first term: The first term in the velocity function is $ t^{0.8} $. We can integrate this term using the power rule of integration, which states that:

tndt=tn+1n+1+C \int t^n dt = \frac{t^{n+1}}{n+1} + C

where $ n $ is a constant, and $ C $ is the constant of integration.

Applying this rule to the first term, we get:

t0.8dt=t1.81.8+C1 \int t^{0.8} dt = \frac{t^{1.8}}{1.8} + C_1

where $ C_1 $ is the constant of integration.

  1. Integrate the second term: The second term in the velocity function is $ 5 \sin (3t - 1) $. We can integrate this term using the substitution method, which involves substituting a new variable into the integral.

Let $ u = 3t - 1 $. Then, $ du = 3 dt $, and $ dt = \frac{du}{3} $.

Substituting these expressions into the integral, we get:

5sin(3t1)dt=53sinudu \int 5 \sin (3t - 1) dt = \frac{5}{3} \int \sin u du

Evaluating the integral, we get:

53sinudu=53cosu+C2 \frac{5}{3} \int \sin u du = -\frac{5}{3} \cos u + C_2

where $ C_2 $ is the constant of integration.

  1. Combine the results: Now that we have integrated both terms, we can combine the results to get the final expression for the position of the particle:

x(t)=t1.81.853cos(3t1)+C x(t) = \frac{t^{1.8}}{1.8} - \frac{5}{3} \cos (3t - 1) + C

where $ C $ is the constant of integration.

Applying the Initial Condition

We are given that the position of the particle is $ x = -5 $ when $ t = 1.5 $. We can use this initial condition to find the value of the constant of integration $ C $.

Substituting $ x = -5 $ and $ t = 1.5 $ into the expression for the position of the particle, we get:

5=(1.5)1.81.853cos(3(1.5)1)+C -5 = \frac{(1.5)^{1.8}}{1.8} - \frac{5}{3} \cos (3(1.5) - 1) + C

Simplifying the expression, we get:

5=2.1971.853cos(4.51)+C -5 = \frac{2.197}{1.8} - \frac{5}{3} \cos (4.5 - 1) + C

5=1.220553cos3.5+C -5 = 1.2205 - \frac{5}{3} \cos 3.5 + C

5=1.22051.262+C -5 = 1.2205 - 1.262 + C

5=0.0415+C -5 = -0.0415 + C

C=5+0.0415 C = -5 + 0.0415

C=4.9585 C = -4.9585

The Final Expression for the Position of the Particle

Now that we have found the value of the constant of integration $ C $, we can substitute it back into the expression for the position of the particle:

x(t)=t1.81.853cos(3t1)4.9585 x(t) = \frac{t^{1.8}}{1.8} - \frac{5}{3} \cos (3t - 1) - 4.9585

This is the final expression for the position of the particle as a function of time $ t $.

Conclusion

In this article, we have explored how to calculate the position of a particle given its velocity function. We used the velocity function $ v(t) = t^{0.8} + 5 \sin (3t - 1) $ and the initial position $ x = -5 $ at time $ t = 1.5 $ to find the position of the particle at any time $ t $. We integrated the velocity function to get the expression for the position of the particle, and then applied the initial condition to find the value of the constant of integration. The final expression for the position of the particle is $ x(t) = \frac{t^{1.8}}{1.8} - \frac{5}{3} \cos (3t - 1) - 4.9585 $.

Introduction

In our previous article, we explored how to calculate the position of a particle given its velocity function. We used the velocity function $ v(t) = t^{0.8} + 5 \sin (3t - 1) $ and the initial position $ x = -5 $ at time $ t = 1.5 $ to find the position of the particle at any time $ t $. In this article, we will answer some common questions related to the motion of the particle.

Q: What is the significance of the velocity function in determining the position of the particle?

A: The velocity function is a crucial component in determining the position of the particle. It describes the rate of change of the particle's position with respect to time. By integrating the velocity function, we can obtain the expression for the position of the particle as a function of time.

Q: How do we handle the constant of integration in the expression for the position of the particle?

A: The constant of integration is a value that is added to the integral to make it exact. In our previous article, we found the value of the constant of integration by applying the initial condition. The constant of integration is essential in ensuring that the expression for the position of the particle is accurate.

Q: What is the role of the initial condition in determining the position of the particle?

A: The initial condition is a critical component in determining the position of the particle. It provides the necessary information to find the value of the constant of integration, which is essential in ensuring that the expression for the position of the particle is accurate.

Q: Can we use the expression for the position of the particle to predict the future position of the particle?

A: Yes, we can use the expression for the position of the particle to predict the future position of the particle. By substituting the desired time into the expression, we can obtain the position of the particle at that time.

Q: How do we handle the trigonometric function in the expression for the position of the particle?

A: The trigonometric function in the expression for the position of the particle is the cosine function. We can handle this function by using the properties of the cosine function, such as the identity $ \cos (a + b) = \cos a \cos b - \sin a \sin b $.

Q: Can we use the expression for the position of the particle to determine the velocity of the particle?

A: Yes, we can use the expression for the position of the particle to determine the velocity of the particle. By differentiating the expression for the position of the particle with respect to time, we can obtain the velocity function.

Q: What is the physical significance of the position of the particle?

A: The position of the particle is a measure of its location in space. It is an essential component in understanding the motion of the particle and predicting its future position.

Q: Can we use the expression for the position of the particle to determine the acceleration of the particle?

A: Yes, we can use the expression for the position of the particle to determine the acceleration of the particle. By differentiating the velocity function with respect to time, we can obtain the acceleration function.

Conclusion

In this article, we have answered some common questions related to the motion of a particle along the x-axis. We have discussed the significance of the velocity function, the role of the initial condition, and the physical significance of the position of the particle. We have also provided examples of how to use the expression for the position of the particle to predict the future position of the particle, determine the velocity and acceleration of the particle, and handle the trigonometric function in the expression.

Additional Resources

For more information on the motion of particles and the use of velocity and position functions, please refer to the following resources:

  • [1] "Motion of Particles" by [Author], [Publisher], [Year]
  • [2] "Velocity and Position Functions" by [Author], [Publisher], [Year]
  • [3] "Trigonometric Functions" by [Author], [Publisher], [Year]

Note: The references provided are fictional and for demonstration purposes only.