Question

A particle moves along the x-axis in such a way that its coordinate $x$ varies with time $t$ according to the expression, $x = 2 - 5t + 1.25{t^2}$. The initial velocity of the particle is?
A. $- 5\,m{s^{ - 1}}$
B. $- 3\,m{s^{ - 1}}$
C. $3\,m{s^{ - 1}}$
D. $6\,m{s^{ - 1}}$

Answer 1

Here, we will calculate the initial velocity of the particle by using the formula of velocity that is the first derivative of the position of the particle with respect to the time. Here, we will take time $t = 0$ because at the initial velocity there will be no displacement of the object.

Complete step by step answer:
It is given in the question that the particle is moving along the X-axis such that the $x$ coordinate varies with time $t$ according to the expression given below,
$x = 2 - 5t + 1.25{t^2}$
Where $x$ will be measured in meters and $t$ will be measured in seconds.
Now, the velocity can be defined as the rate of change of position of the particle with respect to the time. Also, the velocity can be calculated as the first derivative of the position of the particle with respect to time as given below
$v = \dfrac{{dx}}{{dt}}$
Now, putting the value of $x$ in the above equation, we get
$v = \dfrac{{d\left( {2 - 5t + 1.25{t^2}} \right)}}{{dt}}$
$\Rightarrow v = - 5 + 2 \times 1.25t$
$\Rightarrow v = - 5 + 2.5t$
Now, when the particle will be in the initial velocity, then time $t = 0$. Therefore, putting time $t = 0$ in the above equation, we get,
$\therefore v = - 5\,m{s^{ - 1}}$
This is the required velocity. Therefore, the initial velocity of the particle is $- 5\,m{s^{ - 1}}$.

Hence, option A is the correct option.

Note: Initial velocity is the velocity that describes how fast an object can travel when the gravity will first apply force on the object. On the other hand, the final velocity is the velocity that is used to measure the speed and direction of an object after it will reach its maximum acceleration. Also, we can say that the initial velocity is the velocity which an object attains at the beginning of the given time period while the final velocity is the velocity which an object attains at the end of the given time period.