(A) A partielc is moving in three dimensions. Its positon...

(A) A partielc is moving in three dimensions. Its positon vector is given by $\vec{r}-6 \hat{i} \mid(3 \mid 4 t) \hat{j} \quad\left(3 \mid 2 t, t^{2}\right) \hat{k}$ Distance are in metres, and the time, $t$, in scconds. (a) What is the velocity vector at $t=+3 \mathrm{sec}$ ? (b) What is the specd (in m/s) at $t=+3$ scc? (c) What is the acceleration vector and what is its magnitude (in $\mathrm{m} / \mathrm{s}^{2}$ ) at $t=+3$ scc? (B) Now, the particle is moving only along the $z$ -axis, and its positon is given by, $\left(\begin{array}{lll}t^{2} & 2 t & 3) \hat{k}\end{array}\right.$ at what time does the particle stand still?

(A) A partielc is moving in three dimensions. Its positon vector is given by $\vec{r}-6 \hat{i} \mid(3 \mid 4 t) \hat{j} \quad\left(3 \mid 2 t, t^{2}\right) \hat{k}$
Distance are in metres, and the time, $t$, in scconds.
(a) What is the velocity vector at $t=+3 \mathrm{sec}$ ?
(b) What is the specd (in m/s) at $t=+3$ scc?
(c) What is the acceleration vector and what is its magnitude (in $\mathrm{m} / \mathrm{s}^{2}$ ) at $t=+3$ scc?
(B) Now, the particle is moving only along the $z$ -axis, and its positon is given by, $\left(\begin{array}{lll}t^{2} & 2 t & 3) \hat{k}\end{array}\right.$
at what time does the particle stand still?

Key Concepts

Motion Along a Specific Axis

When a particle is constrained to move along a specific axis, such as the z-axis, its position, velocity, and acceleration involve only the component along that axis. Understanding motion in one dimension simplifies the analysis since only one component is non-zero.

Static Condition (At Rest)

A particle is said to be at rest when its velocity is zero. Identifying the time at which the velocity becomes zero is important for understanding moments of instantaneous rest in the motion. For motion along a particular axis, this condition is determined by setting the derivative of the position (the velocity) to zero and solving for time.

Speed Calculation

Speed is defined as the magnitude of the velocity vector. It is a scalar quantity that measures how fast a particle is moving regardless of its direction. The calculation involves taking the square root of the sum of the squares of the velocity components.

Acceleration Vector

The acceleration vector is the derivative of the velocity vector with respect to time. It describes how the velocity of the particle changes over time, providing information about the rate of change of the speed and direction. In particle motion, this vector is crucial for understanding dynamic behavior such as speeding up, slowing down, or changing direction.

Velocity Vector

The velocity vector is the derivative of the position vector with respect to time. It not only indicates the speed of the particle but also the direction of motion. In three dimensions, obtaining the velocity vector involves differentiating each component of the position vector separately with respect to time.

Position Vector

A position vector represents the location of a point in space relative to an origin. In three-dimensional motion, it is typically expressed in terms of its x, y, and z components, which can be functions of time. The position vector is fundamental in describing the trajectory of a particle in space.

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