The radius vector of a point A relative to the origin varies...

The radius vector of a point $A$ relative to the origin varies with time $t$ as $\vec{r}-a t \hat{i}-b t^{2} \hat{j} ;$ where $a$ and $b$ arc positive constants. Find: (a) equation of the point's trajectory $y(x)$, (b) lime dependence of velocity $\vec{v}$ and acceleration $\bar{f}$ and their moduli, (c) lime dependence of the angle $\theta$ bctween $\bar{v}$ and $\vec{f}$, and (d) mean velocity vector averaged over the first $t$ scconds of motion and its modulus.

The radius vector of a point $A$ relative to the origin varies with time $t$ as $\vec{r}-a t \hat{i}-b t^{2} \hat{j} ;$ where $a$ and $b$ arc positive constants. Find:
(a) equation of the point's trajectory $y(x)$,
(b) lime dependence of velocity $\vec{v}$ and acceleration $\bar{f}$ and their moduli,
(c) lime dependence of the angle $\theta$ bctween $\bar{v}$ and $\vec{f}$, and
(d) mean velocity vector averaged over the first $t$ scconds of motion and its modulus.

Key Concepts

Parametric Representation and Elimination of the Parameter

A particle’s motion in a plane is often described by parametric equations where both x and y coordinates are expressed as functions of a third variable, typically time. Eliminating the parameter to obtain a direct relationship between x and y yields the trajectory or path of the particle. This process is fundamental in converting time?dependent motion equations into a spatial curve, which can simplify the analysis of the particle’s path.

Velocity and Acceleration in Kinematics

Velocity is defined as the time derivative of the position vector, and acceleration is the time derivative of the velocity vector. These kinematic quantities provide insight into both the speed and the directional behavior of a moving particle. Understanding how to differentiate vector functions to obtain velocity and acceleration is key to describing and predicting the motion in classical mechanics.

Vector Magnitude (Norm) Computation

The magnitude or norm of a vector is computed using the square root of the sum of the squares of its components. This calculation is essential in kinematics to determine the scalar speed from a velocity vector or the absolute value of acceleration. It provides a measure of how ‘big’ a vector is regardless of its direction.

Dot Product and Angle Between Vectors

The dot product is a fundamental operation in vector algebra used to quantify the projection of one vector onto another. It plays a crucial role in calculating the angle between two vectors by relating the dot product to the product of the magnitudes of the vectors and the cosine of the angle between them. This concept is particularly important in analyzing how the direction of velocity compares to the direction of acceleration.

Average Velocity Vector

The average velocity vector over a given time interval is defined as the net displacement divided by the time interval. This concept offers a simplified overview of the motion by representing the cumulative effect of the instantaneous velocities, and its computation is important for understanding overall motion trends in kinematics.

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