Animated Solution for Physics - Kinematics: Components along and perpendicular to a position vector are known as radial and transverse components respectively. A particle is projected horizontally from the top a tower. Assume acceleration due to gravity to be uniform and the origin of the coordinate system at the point of projection.

Unraveling the Secrets of Radial and Transverse Components

Imagine standing at the edge of a towering cliff, a stone in your hand. You toss it perfectly horizontally into the void. As it traces a graceful parabolic arc towards the ground, its velocity and acceleration are constantly changing. But what if we view this motion not through the rigid, static lens of standard $x$ and $y$ axes, but through a dynamic, rotating frame of reference that tracks the stone's every move? Welcome to the fascinating world of radial and transverse components.

The Coordinate System

To make sense of the chaos, we first need an anchor. The problem explicitly instructs us to place the origin of our coordinate system exactly at the point of projection. Let's define the horizontal direction as our $x$-axis and the vertically downward direction as our positive $y$-axis.

Since the stone is projected horizontally with an initial speed $u$, its horizontal position at any time $t$ is simply $x=ut$. Vertically, it falls under the influence of gravity, so its position is $y=\frac{1}{2}g{t}^2$.

This gives us our master position vector:

Radial and Transverse Axes

Now, let's introduce our dynamic axes. The radial direction (denoted by the unit vector ${\hat{e}}_r$) is the line pointing straight from the origin to the stone's current position. The transverse direction (${\hat{e}}_{\theta }$) is exactly perpendicular to it, pointing in the direction that the angle $\theta$ is increasing.

What is this angle $\theta$? It's the angle the position vector makes with the horizontal $x$-axis. Using basic trigonometry, we can find it:

As the stone falls, time $t$ increases, which means $\tan \theta$ increases. Consequently, the angle $\theta$ continuously grows from $0$ towards $\pi /2$ (or $90^{\circ }$). This continuous growth of $\theta$ is the engine that drives the behavior of our components.

The velocity vector $\vec{v}$ is always tangent to the trajectory. By differentiating our position vector, we get:

To find the radial velocity $v_r$, we project this velocity vector onto the radial axis using a dot product ($v_r=\vec{v}\cdot {\hat{e}}_r$). Geometrically, this yields:

Here is where the magic happens. We know from earlier that $gt=2u\tan \theta$. Substituting this into our equation gives:

Look closely at this beautiful fraction. As the stone falls, $\theta$ increases. The numerator contains ${\sin }^2\theta$, which grows larger. The denominator is $\cos \theta$, which shrinks smaller. A growing numerator divided by a shrinking denominator means the entire fraction explodes upwards. Thus, the radial velocity always increases.

What about the transverse velocity $v_{\theta }$? Projecting the velocity onto the transverse axis gives:

Again, we substitute $gt=2u\tan \theta$:

The complex expression collapses into a stunningly simple result! Since $\theta$ is always increasing as the stone falls, $\sin \theta$ is also always increasing. Therefore, the transverse velocity always increases.

Acceleration is much simpler. The only force acting on the stone is gravity, so the total acceleration vector is a constant $\vec{a}=g\hat{j}$ pointing straight down.

When we project this constant downward vector onto our rotating axes, the geometry is straightforward. The radial acceleration is the component of gravity pulling along the position vector:

Since $\theta$ is increasing, $\sin \theta$ is increasing. Thus, the radial acceleration always increases.

The transverse acceleration is the component of gravity acting perpendicular to the position vector:

As $\theta$ increases, $\cos \theta$ decreases. Therefore, the transverse acceleration always decreases.

By systematically breaking down the motion into this rotating frame, we've uncovered a hidden symphony of increasing and decreasing components.

- Radial velocity ($v_r$) always increases. - Transverse velocity ($v_{\theta }$) always increases. - Radial acceleration ($a_r$) always increases. - Transverse acceleration ($a_{\theta }$) always decreases.

Matching these findings with the given columns provides the flawless solution to this matrix match challenge. The next time you see a projectile, don't just look at its $x$ and $y$ coordinates—try to visualize the invisible, rotating radial and transverse axes dancing along with it!