Explanation of Particle Motion along a Straight Line

The position of a particle moving along a straight line can be described by a position function. In this case, the position function of the particle is given by:

x = t3 - 3t2 + 3t + 3

This function describes the position of the particle at any given time t.

Deriving the Velocity Function

To understand the motion of the particle, we need to derive its velocity function. The velocity function is the derivative of the position function with respect to time:

v = dx/dt = 3t2 - 6t + 3

This function describes the velocity of the particle at any given time t. We can use this function to determine the direction and speed of the particle.

Deriving the Acceleration Function

To further understand the motion of the particle, we need to derive its acceleration function. The acceleration function is the derivative of the velocity function with respect to time:

a = dv/dt = 6t - 6

This function describes the acceleration of the particle at any given time t. We can use this function to determine if the particle is accelerating or decelerating, and at what rate.

Interpreting the Results

Using the velocity and acceleration functions, we can interpret the motion of the particle:

- The particle is moving in the positive direction when its velocity is positive, and in the negative direction when its velocity is negative.
- The particle is accelerating in the positive direction when its acceleration is positive, and decelerating in the positive direction when its acceleration is negative.
- The particle is at rest when its velocity is zero, and its acceleration is also zero at this point.
- The particle reaches a maximum height or distance when its velocity is zero and its acceleration is negative.
- The particle reaches a minimum height or distance when its velocity is zero and its acceleration is positive.

Conclusion

In conclusion, the position function, velocity function, and acceleration function provide a complete understanding of the motion of a particle along a straight line. By analyzing these functions, we can determine the direction, speed, and acceleration of the particle at any given time.