Question

If the particle is moving along a straight line given by the relation $x=2-3t+4{{t}^{3}}$ where s is in cms., and t in sec. Its average velocity during the third sec is then
$\begin{aligned} & \left( A \right)105cm/s \\\ & \left( B \right)80cm/s \\\ & \left( C \right)85cm/s \\\ & \left( D \right)90cm/s \\\ \end{aligned}$

Answer 1

Here the relation for displacement, x is given. Hence the average velocity is the rate of change of displacement with time. Then substitute the value of t in the resultant equation obtained by differentiating the given equation for displacement with respect to time t. Thus we will get the average velocity in 3s.
Formula used:
Average velocity$=\dfrac{Displacement}{Time}$
$V=\dfrac{dx}{dt}$
V is the average velocity
dx is the displacement
and dt is the change in time.

Complete step by step solution:
Given that, $x=2-3t+4{{t}^{3}}$ ………(1)
Average velocity$=\dfrac{Displacement}{Time}$
Differentiating the given equation (1) for displacement with respect to time t. Thus we will get the average velocity ,
$\Rightarrow \dfrac{dx}{dt}=-3+12{{t}^{2}}$
That is $V=\dfrac{dx}{dt}$
$\Rightarrow V=-3+12{{t}^{2}}$
At time t=3s
The average velocity becomes,
$\begin{aligned} & \Rightarrow V=-3+12\times {{\left( 3 \right)}^{2}}^{_{{}}} \\\ & \Rightarrow V=105cm/s \\\ & \\\ \end{aligned}$

Therefore option (A) is correct.

Additional information:
The average velocity has the same SI unit of velocity and is m/s. It is a vector quantity. Where the velocity can be described as the rate in which the position changes. The average velocity is a vector quantity since the displacement is a vector quantity. Thus displacement can be described as the change in object’s position from one place to another. That is, the change in overall position of the object. But distance is the true path and hence a scalar quantity.

Note:
The average velocity can be defined as the ratio of total displacement to the total time taken. Otherwise average velocity can also be defined as the ratio of rate of change of displacement to the change in time. The average velocity has the same SI unit of velocity and is m/s. It is a vector quantity. Where the velocity can be described as the rate in which the position changes. The average velocity is a vector quantity since the displacement is a vector quantity. Thus displacement can be described as the change in object’s position from one place to another. That is, the change in overall position of the object. But distance is the true path and hence a scalar quantity.