Question

The displacement-time equation of a particle moving along the x-axis is$x={{t}^{2}}-5t+45$ where x is in meters and t is in seconds. The velocity of the particle at time t=0 is
A. -5m/s
B. 12m/s
C. 5m/s
D. -12m/s

Answer 1

Recall the definition of velocity. Note that we are asked to find the velocity at a particular instant, that is, the instantaneous velocity at t =0. We are given the displacement-time equation and we could differentiate that with respect to time to get the expression of instantaneous velocity in terms of time t. Now you could substitute t =0 to get value.

Formula used:
Expression for instantaneous velocity,
$v=\dfrac{dx}{dt}$

Complete step by step answer:
We are given the displacement time equation of a particle moving on the x-axis as,
$x={{t}^{2}}-5t+45$ …………………….. (1)
Where x is the displacement in meters and t is the time in seconds. We are asked to find the velocity of the particle when time t=0.
Let us recall the definition of velocity.
We know that displacement of a body is the difference in final and initial positions during the given time interval. Velocity is the time rate of change of displacement of a body. Average velocity can tell you how fast the body is moving over a particular interval of time and is given by,
$\overline{v}=\dfrac{x}{t}$
But by knowing the average velocity we cannot know the velocity at a particular instant of time, for that we need instantaneous velocity (v). Instantaneous velocity can be defined as the limit of the average velocity as the time interval $\Delta t$ becomes infinitely small. That is,
$v=\underset{\Delta t\to 0}{\mathop{\lim }}\,\dfrac{\Delta x}{\Delta t}$
In calculus, the R.H.S of this equation is the differential coefficient of x with respect to t and is denoted by,
$v=\dfrac{dx}{dt}$ ……………………….. (2)
Now we know that differentiating (1) with respect to t will give us the expression for instantaneous velocity.
$\dfrac{dx}{dt}=2t-5$
From (2),
$v=2t-5$
Substituting t = 0,
$v=2\left( 0 \right)-5=-5m{{s}^{-1}}$
Therefore the velocity of the body at the instant t =0 is -5m/s.
Hence the answer to the question is option A.

Note:
We could also obtain the value of velocity at an instant graphically other than this numerical method. We can determine the instantaneous velocity from the position-time graph. If we were to obtain the velocity at t=0s graphically, all you have to do is find the slope of the tangent to that graph at t=0 and this slope gives you the velocity at that instant.