Question

A particle is moving with speed $v = b\sqrt x $ along positive x-axis. Calculate the speed of a particle at time $ t = \tau $. (Assume that the particle is at origin at t=0).

Answer 1

Hint: Here, in this question we know that the acceleration is the rate of change of the velocity of an object with respect to time. By putting the given value of velocity in the acceleration, we get the resultant speed of the particle at time t.

Formula used:
$\dfrac{{dv}}{{dt}} = \dfrac{b}{{2\sqrt x }}\dfrac{{dx}}{{dt}}$

Complete step-by-step answer:
We know that the acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is a vector quantity. The unit of acceleration is m/s2.
Here we are given
$v = b\sqrt x $
Now, by differentiating the above equation of velocity with respect to time t, we get:
$\dfrac{{dv}}{{dt}} = \dfrac{b}{{2\sqrt x }}\dfrac{{dx}}{{dt}}......(1)$
As we already know that, rate of change of velocity is called acceleration:
$\dfrac{{dv}}{{dt}} = a$
We also know that velocity is given by the rate of change of position of any object:
$\dfrac{{dx}}{{dt}} = v$
Now, by substituting the value of acceleration a and velocity v in equation (1) and solving, we get:
$a = \dfrac{{bv}}{{2\sqrt x }}$
Now, by putting the given value of v in the above equation we get:
$a = \dfrac{{{b^2}}}{2}$
Or,
$\dfrac{{dv}}{{dt}} = \dfrac{{{b^2}}}{2}$
Now, to get the speed or the velocity v at the time$t = \tau $, we integrate both sides with respect to time, we get:
$\int {\dfrac{{dv}}{{dt}}} = \int {\dfrac{{{b^2}}}{2}} $
$\therefore v = \dfrac{{{b^2}}}{2}\tau $
Therefore, assuming that the particle is at origin at t=0.the speed of a particle at time $t = \tau $ is given by v.

Additional Information: Speed is a scalar quantity, which means it has magnitude only. The rate at which an object covers distance per time, whereas, velocity is a vector quantity this means it has both magnitude and direction. Velocity of an object is given by the rate of change of its position with respect to a frame of reference. It is a function of time. The unit of velocity is m/sec.
Time is defined by its measurement. Time is what a clock reads. In classical and non- realistic 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.

Note: If velocity of a particle is constant then its acceleration will be zero, because differentiation of a constant value is zero. As acceleration results in increase of speed whereas, deceleration results in decrease of speed or velocity. In 1-D the deceleration is defined as the acceleration which is opposite to the velocity.