A Simple Note on Instantaneous Velocity

Instantaneous velocity, sometimes known as just velocity, is the number that informs us how quickly an item is travelling somewhere along its route. If two places on a path are separated by a significant amount of time (and hence by a large amount of distance), this is the average velocity between them. It is necessary to describe location x as a continuous function of time, indicated by the symbol x(t). With this representation, the equation for the average velocity between two places is

v = x(t2) – x(t1)t2 – t1 

Instantaneous Velocity

The instantaneous velocity at any point in time may be determined by using the following equation:

t1 = t and t2 = t + Δt

At time t1 = t position is defined as x(t1) = x

and at time  t2 = t+ Δt, the position is defined as x(t2) = x+ Δx

After incorporating these expressions into the equation for average velocity and determining the upper and lower limits, we have

 Δt→0

 as a result, we obtain the following formula for the instantaneous velocity:

vinstant= limt →0 xt

 vinstant = dxdt

For example, the speedometer in your automobile informs you how fast you are travelling at any one time when driving. If we want to know the momentum of each item just before a collision, we may look at the instantaneous velocity of each object.

Formula For Instantaneous Velocity

Instantaneous Velocity is calculated using the following formula:

It is determined similarly to average velocity, but the period is significantly shorter. Remember that the average velocity for a particular period is the total displacement divided by the whole time, which we learned before. Because this time interval is approaching 0, the displacement is also zero. Nevertheless, the maximum value of this displacement ratio to time is not zero; this is called instantaneous velocity. 

Instantaneous Acceleration The following formula may be used to represent the given body at any given point in time: 

vinstant= limt →0 xt= dxdt

In which case, x is the function specified about time t. The Instantaneous Velocity is measured in metres per second (m/s). Vint=The body’s instantaneous velocity is measured in metres per second. 

t is the short period between events. 

The letter x denotes the displacement variable.

Graphical Representation

By drawing a graph between position and time, we can find velocity because velocity is defined as the slope of the position-time graph. A position-time graph can assist you in thinking about average and instantaneous velocity in a more organised manner than you would otherwise. As a result, the average velocity is provided by the change in position divided by the length of time it takes to accomplish that change. In contrast, the instantaneous velocity is a slight change in position in minimal time divided by this short time interval. 

Instantaneous Velocity Calculation

If we want to calculate instantaneous velocity, we must first define the explicit version of the position function of time, i.e. x(t). Suppose that each term in the x(t) equation takes Atn, where A is a constant and n is an integer. Using the power rule, we may differentiate each term to give the following:

 dxdt= d(Atn)dt

dxdt= nAtn-1

Numericals on instantaneous velocity

Q1: The displacement of a particle is provided by the expression x(t) = 10 t2 – 5t + 1. Calculate the instantaneous velocity of the object at time t = 3s.

Solution : Instantaneous velocity is given by 

v = t0 st

   =t0 (x(t+t))- x(t)t

   = t3 d(10t2 – 5t +1)dt

   = t3 (20t – 5)

Speed(i) =(20(3)-5)

Speed(i) = 60-5

Speed(i) = 55m/s

Q2. Using a position function x defined as 5t2 + 2t + 4 and a time interval of 3 seconds, what is the instantaneous velocity of an object going straight ahead for 3 seconds? 

Solution :  According to the question position function is,

  x = 5t2 + 2t + 4

 We compute Instantaneous Velocity by differentiating the given function concerning t in the following way:

 vinstant = dxdt

Substituting function x,

  vinstant = 10t + 2

Put the value of t= 3, and we get the instantaneous velocity as,

 vinstant =10 3 +2

 vinstant = 32 m/s

As a result, the instantaneous velocity of the primary function is 32 m/s.

Conclusion

Instantaneous velocity is a vector quantity used to find out the velocity of an object in motion at a particular point of time when a certain time interval is taken into account. It is an important modality which can be used by physicists, mathematicians, statisticians to solve problems related to our day to day lives and understand the physics behind it. Derivation of the formula for instantaneous velocity can be made via calculus or a graphical method. Though the graphical method is a bit difficult, the mathematical method is widely used and easier to perform. Instantaneous velocity forms an integral part of kinematics.