(a) If a particle's position is given by x=4-12 t+3 t^2...

(a) If a particle's position is given by $x=4-12 t+3 t^{2}$ (where $t$ is in seconds and $x$ is in meters), what is its velocity at $t=1 \mathrm{~s} ?(\mathrm{~b})$ Is it moving in the positive or negative direction of $x$ just then? (c) What is its speed just then? (d) Is the speed increasing or decreasing just then? (Try answering the next two questions without further calculation.) (e) Is there ever an instant when the velocity is zero? If so, give the time $t ;$ if not, answer no. (f) Is there a time after $t=3 \mathrm{~s}$ when the particle is moving in the negative direction of $x ?$ If so, give the time $t ;$ if not, answer no.

(a) If a particle's position is given by $x=4-12 t+3 t^{2}$ (where $t$ is in seconds and $x$ is in meters), what is its velocity at $t=1 \mathrm{~s} ?(\mathrm{~b})$ Is it moving in the positive or negative direction of $x$ just then? (c) What is its speed just then? (d) Is the speed increasing or decreasing just then? (Try answering the next two questions without further calculation.) (e) Is there ever an instant when the velocity is zero? If so, give the time $t ;$ if not, answer no.
(f) Is there a time after $t=3 \mathrm{~s}$ when the particle is moving in the negative direction of $x ?$ If so, give the time $t ;$ if not, answer no.

Key Concepts

Kinematic Equations

Kinematics describes the motion of objects using position, velocity, and acceleration as functions of time. The position function gives the location, while its first derivative with respect to time yields the velocity, and the second derivative yields the acceleration. These relationships help analyze an object’s motion in a systematic way.

Velocity as the Derivative of Position

In calculus-based physics, velocity is defined as the time derivative of the position function. By differentiating the position equation with respect to time, you obtain a velocity function that provides the instantaneous rate of change of position, which is critical for understanding motion at any given moment.

Direction of Motion and Sign of Velocity

The sign of the velocity (positive or negative) indicates the direction in which the particle is moving along a coordinate axis. A positive velocity denotes movement in the positive direction, while a negative velocity denotes movement in the opposite direction. This concept is crucial for analyzing the motion’s orientation.

Speed as the Magnitude of Velocity

Speed is defined as the absolute value of velocity, representing how fast an object is moving regardless of its direction. It is a scalar quantity, meaning only the magnitude is concerned, not the direction, making it fundamental when assessing the intensity of motion.

Acceleration and Change of Speed

Acceleration, the second derivative of the position function, indicates the rate at which velocity changes over time. The relative signs of velocity and acceleration determine whether the speed is increasing or decreasing. If both share the same sign, the speed increases; if they are opposite, the speed decreases.

Zero Velocity and Turning Points

A moment when the velocity equals zero indicates a possible change in the motion’s direction, often referred to as a turning point. Analyzing when and if velocity becomes zero helps identify when the particle might momentarily stop before reversing its direction, a key aspect in understanding the full motion behavior.

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