The vector position of a particle varies in time aceording...

The vector position of a particle varies in time aceording to the expression $\mathbf{r}=\left(3.00 \mathbf{i}-6.00 t^{2} \mathbf{j}\right) \mathrm{m} .$ (a) Find expressions for the velocity and acceleration as functions of time. (b) Determine the particle's position and velocity at $t=1.00 \mathrm{~s}$

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Key Concepts

Vector Representation of Motion

This concept involves describing the position of a particle using a vector function, where each component of the vector corresponds to a coordinate in space. It provides a complete depiction of the particle's location in a multidimensional setting as a function of time.

Differentiation of Vector Functions

In kinematics, finding the velocity and acceleration of a particle requires differentiating its position vector with respect to time. The first derivative yields the velocity, which describes how the particle's position changes over time, and the second derivative gives the acceleration, indicating the rate of change of the velocity.

Evaluating Functions at Specific Time Instants

After obtaining the general expressions for position, velocity, and acceleration, substituting a particular time value into these expressions allows us to determine the instantaneous state of the particle. This is essential for analyzing the motion at any given moment, making it a fundamental tool in physics and engineering.

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