Instantaneous Velocity

When we’re driving in a car, the speedometer displays how fast the vehicle is moving at the time. An object’s speed at one point is called “instantaneous velocity.” The average velocity is correlated with an average rate of change. In the same way, the instantaneous velocity is associated with a rapid change in speed.

Instantaneous velocity refers to the speed at which you change your location over a short period of time that is extremely small. The measurement is made using the SI measurement unit, m/s. Instantaneous speed is similar to instantaneous velocity. It is of the same magnitude as instantaneous velocity. However, it does not have any direction.

In simple terms, the instantaneous velocity is defined as “the velocity of an object under motion at a specific point of time.”

What is instantaneous velocity?

Now let us understand, “what is instantaneous velocity?” If the object is in uniform velocity, then its velocity at the instantaneous moment may be the same as the normal velocity.

It is calculated in a similar way to the average velocity. However, here the time intervals are shortened. We are aware that the average speed at a given time period is the total displacement divided by the total time. As in the case of instantaneous velocity, the time interval is nearing zero, the velocity is close to zero. But, the maximum ratio of time to displacement is not zero, and the value is referred to as instantaneous velocity.

The rate of change in the displacement for an object within a specific direction is known as its velocity. The SI unit is metres per second.

The direction of the instantaneous velocity at any given moment indicates the direction of motion of particles at that moment in time. The amount of instantaneous velocity equals the speed of the instantaneous particles. This is due to the fact that even for a tiny time period, the motion of particles is approximated to be uniform.

We can define instantaneous velocity as the measurement of the speed of the object from one place to another. When the duration and the distance between them are not more than zero, you can calculate the average speed along the path between two points.

Instantaneous velocity formula

To illustrate this concept mathematically, we will derive the instantaneous velocity formula. It can be denoted as x(t), where x is the position and t is the constant function. We can describe the average velocity between two points using the expression:

v = x(t2) – x(t1)/t2 – t1

To determine the velocity instantaneously at any given point, we’ll have t1=t and t2=t+Δt. We use the equation of the average velocity and add these expressions to them by including the limiting factor as:

Δt→0, we can find the expression for the instantaneous speed:

v(t) = Δt0[x(t + Δt) – x(t)]/Δt = dx(t)/dt

The speed of an object is the average velocity at the point when the time that has passed by is close to zero and is presented as the derivation of x relative to the time t.

Average velocity and instantaneous velocity

A person’s average speed can be calculated by dividing its total displacement by the time required to move from one spot to the next.

Average velocity is the change of location/change in time, v = d/t

Average velocity does not reveal the exact speed at which an object moves at each and every second of the time. The object may accelerate, slow down, or even be stopped for a short time, but this won’t significantly affect its average speed. 

Let’s take an example.

When it takes Jack an average of 30 minutes to travel eight miles from the house to school, what was his average speed?

Vav = d / t = (8 miles ) / (30 minutes) = 0.27 miles/min

In contrast, instantaneous velocity refers to the speed of the object at one moment in time. This could or could not be exactly the same as the average speed over a longer duration. In the case of Jack, when he’s waiting to see the train go by, his speed is zero since his car isn’t moving in any way. However, his average speed throughout the whole trip to school won’t be zero, even though his speed was zero in only a tiny portion of the journey.

The instantaneous velocity for a particular time point t0 represents the rate of change in the function of position that corresponds to function x(t) slope at t0. 

The velocity at the moment of measurement is shown at the time t0. This is the highest point of the function position, which will state that the instantaneous velocity is zero as the slope of the graph of the position will be zero at this time. 

The instantaneous velocity is not calculated as zero during the rest of the times, such as t1,t2, and so on. That’s because, in that case, the slope on the graph could be both negative and positive. If the function for the position had at least a minimum, then its slope in the graph would also be zero, which would result in an instantaneous velocity of zero too. Therefore, the zeros of the velocity function are the maximum and the minimum function of position.

Conclusion

The instantaneous velocity is the speed of change in position over a duration that is very low, i.e., almost zero. It is measured in SI units as ms-1. Additionally, the magnitude of instantaneous velocity is the same as instantaneous speed, but with the exception that instantaneous speed doesn’t have any direction.