Question

The position vector of a particle is given by $\vec r = \vec r{}_0(1 - at)t$, where t is the time and a as well as $r{}_0$ are constant. After what time the particle will return to the starting point?

Answer 1

Hint: In this question, we have the position vector of a particle having acceleration a and time t. further by substituting the conditions given in the question, we will get the required result. Also, we will discuss the basics of time, for our better understanding.
Formula used:
$\vec r = \vec r{}_0(1 - at)t$

Complete answer:
Here, we have the position vector of a particle, which is given as:
$\vec r = \vec r{}_0(1 - at)t$
We have initial conditions:
$t = 0,\vec r = 0$
After completing one rotation, the position vector becomes zero, and our equation becomes:
$\vec r = \vec r{}_0(1 - at)t$
$ \Rightarrow 0 = \vec r{}_0(1 - at)t$
$ \Rightarrow 0 = \vec r{}_0t - {\vec r_0}at{}^2$
Now, solving for this quadratic equation, we get:
$t = 0,\dfrac{1}{a}$
$\therefore t = \dfrac{1}{a}$
Therefore we get the required answer, which shows at this time t, the particle will return to its starting point.

Additional information:
As we know, time is defined by its measurement. Time is what a clock reads. In classical and non- relativistic physics, it is a scalar quantity. We know that length, mass and charge is usually described as a fundamental quantity, similarly time is also a fundamental quantity. The S.I unit of time is second.
As we know that energy is defined as the ability to do work. Energy can be found in many things and can take different forms, like-kinetic energy is the energy of motion, and potential energy is energy due to an object's position or structure. Total energy of a system is given by the sum of kinetic energy and potential energy.
We also know that, according to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another. Here, in this question electrical energy is transferred to heat energy.
We should also know about the power. It is the amount of energy transferred or converted per unit time. In the International System or S.I Unit of power is the watt. The S.I unit of power can also be written as one joule per second. Power is a scalar quantity, which means it has only magnitude not direction.

Note:
We should remember the S.I unit of time is second, which is represented s. One minute is equal to 60 seconds and 1 hour is equal to 3600 sec. Also, the laws of conservation of energy can never be violated in any case. According to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another