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When we talk about particle motion, we aim to understand how an object moves along a path over a given period. In this case, the particle moves along the x-axis, which is a one-dimensional path, making it easier to analyze. The motion of this particle is dictated by a specific mathematical equation:
\[ x = c t^2 - b t^3 \]Here, \( x \) represents the position of the particle in meters, \( t \) is the time in seconds, and \( c \) and \( b \) are constants that shape the particle's path. The equation is quadratic and cubic, indicating it can have curves, slopes, and turning points. This suggests the particle's speed does not remain constant but changes as time progresses. Understanding the fundamentals of particle motion helps us predict where the object will be in the future or where it has been, based on the time elapsed.

Velocity is a fundamental concept in kinematics and refers to the rate of change of the particle's position. If you think about it, velocity tells you how fast something is moving and in what direction. For our particle, the velocity can be found by taking the derivative of the position function with respect to time:
\[ v(t) = \frac{dx}{dt} = 2ct - 3bt^2 \]This formula indicates how the velocity changes as time passes. Significantly, the velocity is dependent on time, meaning its speed and direction vary. By solving this equation at different time points, we can determine the particle's velocity.

• At \( t = 1 \text{ s} \), \( v(1) = 2 \text{ m/s} \)
• At \( t = 2 \text{ s} \), \( v(2) = -8 \text{ m/s} \)
• At \( t = 3 \text{ s} \), \( v(3) = -30 \text{ m/s} \)
• At \( t = 4 \text{ s} \), \( v(4) = -64 \text{ m/s} \)

Notice the switch from positive to negative values, suggesting the particle changes direction.

Acceleration tells us how the velocity of the particle changes over time. This is like understanding how our car's speedometer readings change if we press the gas pedal. Acceleration is calculated by taking the derivative of the velocity function:
\[ a(t) = \frac{dv}{dt} = 2c - 6bt \]This expression shows that the acceleration is directly influenced by time and the constants \( c \) and \( b \). This allows us to compute the acceleration at specific moments:

• At \( t = 1 \text{ s} \), \( a(1) = -4 \text{ m/s}^2 \)
• At \( t = 2 \text{ s} \), \( a(2) = -16 \text{ m/s}^2 \)
• At \( t = 3 \text{ s} \), \( a(3) = -28 \text{ m/s}^2 \)
• At \( t = 4 \text{ s} \), \( a(4) = -40 \text{ m/s}^2 \)

The negative values indicate that the particle is decelerating as time increases.

Displacement and distance, while related, describe different properties of the particle's path. Displacement measures the change in the particle's position, essentially the straight path from start to end. On the other hand, distance is the total length of the path traveled, regardless of direction.
To find the displacement between \( t = 0 \text{ s} \) and \( t = 4 \text{ s} \), calculate:
\[ x(4) = 4 \cdot 16 - 2 \cdot 4^3 = 64 - 128 = -64 \text{ m} \]The displacement from \( t = 0 \text{ s} \) to \( t = 4 \text{ s} \) is then \(-64 \text{ m} \).For the distance, consider all segments of motion, including changes in direction. Compute position at the keypoints, find absolute values of all segments:
*At each turning point and endpoints, calculate the total length traveled.*Thus, while the displacement is \(-64 \text{ m} \), indicating a net movement in one direction, the distance traveled could be greater, factoring in every directional change.