Animated Solution for Physics - Kinematics: The coordinates of a moving particle at any time $t$ are given by $x=\alpha t^3$ and $y=\beta t^3$. The speed of the particle at time $t$ is given by

The problem asks us to find the speed of a moving particle whose coordinates are given as functions of time. This is a classic kinematics problem where we need to connect position, velocity, and speed.

Imagine a particle moving in a two-dimensional plane. Its position at any instant $t$ is completely determined by its $x$ and $y$ coordinates:

Notice how both coordinates depend on the cube of time. This tells us that the particle is accelerating, and its velocity is constantly changing. But what does its path look like? If we divide $y$ by $x$, we get $\frac{y}{x}=\frac{\beta }{\alpha }$, which means $y=\left(\frac{\beta }{\alpha }\right)x$. Surprisingly, despite the cubic time dependence, the particle is actually moving in a straight line!

To find the speed, we first need to determine the velocity. Velocity is a vector, and in two dimensions, it has two components: $v_x$ and $v_y$. These components are simply the rates of change of the respective coordinates with respect to time.

Let's calculate the horizontal velocity component, $v_x$, by differentiating $x$ with respect to $t$:

Similarly, the vertical velocity component, $v_y$, is found by differentiating $y$:

Now that we have both components of the velocity vector, we can find the net speed. Speed is the magnitude of the velocity vector. Since $v_x$ and $v_y$ are perpendicular to each other, we can use the Pythagorean theorem:

Let's substitute our calculated components into this formula:

Squaring the terms inside the square root gives:

We can factor out the common term $9t^4$:

Finally, taking $9t^4$ out of the square root yields our final answer:

The speed of the particle increases quadratically with time. This elegant result perfectly captures the dynamics of the particle's motion!