91Ó°ÊÓ

In kinematics, particle motion describes how a particle moves through space over time. Understanding the particle's motion involves knowing the particle's position, velocity, and acceleration at any given time. This is often expressed using mathematical functions.
In this problem, we are focused on a particle that starts at a position of 24 meters when time is zero. By the time 6 seconds have passed, it has moved to a position of 96 meters with a velocity of 18 m/s. We need to determine how its position and velocity change continuously over time.
This requires setting up equations for position and velocity functions. These functions help map out the motion of the particle as it both speeds up and slows down, reacting to the forces acting on it.

Acceleration describes how quickly the velocity of a particle changes over time. In this scenario, the acceleration is directly proportional to the square of time, represented mathematically as \( a = k \cdot t^2 \), where \( k \) is a constant.
This means as time squares and grows larger, the acceleration increases more dramatically. Essentially, the long the time, the greater the acceleration.Knowing this relationship allows us to integrate acceleration to find the velocity function. By doing so, we can understand how the velocity changes at any given instant.
Acceleration is crucial as it gives insight not only into the speed but also the direction of a particle's motion over time.

Velocity is a vector that describes the speed and direction of a particle's motion. For our problem, the velocity function can be found by integrating the acceleration function \( a = k \cdot t^2 \) over time, leading to \( v(t) = \frac{k t^3}{3} + C_v \).
At \( t = 6 \) seconds, we know that the velocity is 18 meters per second. This specific information allows us to solve for constants in our velocity function, like \( k \) and \( C_v \). By doing so, we make the velocity equation unique to this specific particle's scenario. Ultimately, velocity functions provide detailed insights into how the particle's speed and direction change over time.

Position functions tell us where a particle is located at any given time. To find the position function \( x(t) \), we integrate the velocity function \( v(t) \).
This operation gives us \( x(t) = \frac{kt^4}{12} + C_v t + C_x \), where \( C_x \) is another constant determined by the initial condition of the particle's position. Using the condition at \( t = 0 \) that \( x = 24 \), we find \( C_x = 24 \).
In situations like this, initial conditions are crucial as they allow us to find the specific constants, making the position function accurately describe the particle's path. We can then see how our particle moves along a line, curve, or in any trajectory defined by this position function.