The motion of a particle is described by the equation x = a + bt² where a = 15 cm and b = 3 cm. Its instantaneous velocity at time 3 sec will be

To find the instantaneous velocity of the particle, we need to understand the relationship between position and velocity. Velocity is defined as the rate of change of position with respect to time. Mathematically, the instantaneous velocity $v$ is the first derivative of the position $x$ with respect to time $t$:

$v=\frac{dx}{dt}$

The given equation of motion is:
$x=a+b{t}^2$

Differentiating $x$ with respect to $t$:
$v=\frac{d}{dt}(a+b{t}^2)$
Since $a$ is a constant, its derivative is 0. Using the power rule, the derivative of $b{t}^2$ is $2bt$. Therefore, the expression for velocity is:
$v=2bt$

Now, we substitute the given values into the velocity equation:
Given constant $b=3\text{cm/s}{s}^2$ (since units of $b{t}^2$ are cm) and we want to find the velocity at time $t=3\text{sec}$:
$v=2\times 3\times 3$
$v=18\text{cm/sec}$

Thus, the instantaneous velocity of the particle at $t=3\text{sec}$ is $18\text{cm/sec}$.