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Speed is and velocity are two different measurements of motion. While speed is a scalar quantity and has no direction velocity is a vector quantity with a definite direction. Average speed can be defined as the total distance travelled by an object divided by the total time taken for the motion to take place.
Instantaneous speed is the speed recorded at a specific point of time/at a particular instant. On the other hand average velocity implies the change in position of an object (displacement) divided by the time period in which the displacement has taken place in a particular direction.
Definition: Instantaneous velocity can be defined as how fast an object moves when it is in motion at a particular instance during that specific time interval which is taken into account.
Velocity differs from speed as it is a vector quantity i.e. it has a direction component. But speed is a two-dimensional modality without any direction component hence it is known as a scalar. Though the SI Unit of both speed and velocity is metres per second (m/s).
Derivation of Equation:
In order to determine instantaneous velocity, we have to understand the underlying concept.
The velocity at a particular instance is known as the limit of average velocity as the time period ∆t becomes small infinitesimally.
v(inst)=lim x(t+Δt)−x(t)/ Δt = dx(t)/dt
Δt→0
where,
lim(symbol) stands for the operation.
The right hand side depicts the differential coefficient of x with respect to t and thereby denoted by dx .
Therefore , dt depicts the rate of change of position with relation time, at that particular instant.
To obtain the value of velocity at an instant , both numerical and graphical methods can be used by applying the above equation.
(At t = 4 s velocity is the slope of the tangent to the graph at that point of time.)
In order to derive the value of velocity graphically, we need to take a sample to understand it better. Suppose we take time t = 4 s (point P) for the motion of the object, say a car, represented in the above graph. The graph has been redrawn by choosing different scales to perform the calculation.
Let ∆t be equal to 2 seconds at t = 4 s. Then, as per the definition of average velocity, the slope of line joining P1 to P2 ( Fig. 1) provides us with the value of average velocity over the time interval of 3 seconds to 5 seconds.
If we reduce the value of ∆t from 2 seconds to 1 second , the line joining P1 and P2 now becomes Q1 to Q2. The new slope of Q1Q2 gives the value of the average velocity of the time interval 3.5 seconds to 4.5 seconds.
When ∆t → 0 becomes the limit , the line joining P1 to P2 becomes tangential with respect to the position versus time curve at point P and the velocity at t equals to 4 seconds, given by the slope of the tangent at that particular point.
To show this process graphically is very difficult.
If we use the numerical method, the understanding of the limiting process becomes clearer and hence obtaining the velocity value becomes feasible.
In the graph shown in Fig.1, say, x = 0.08 at t3. Table 1 shows us the value of ∆x/∆t when calculated for ∆t equal to 2.0 s, 1.0 s, 0.5 s, 0.1 s and 0.01 s respectively, with t = 4.0 s. The second and third column gives the value of
t1= {t – ∆t/2} and t2= {t + ∆t/2}
The fourth and the fifth column give the corresponding values of x, i.e.
x(t1)=0.08 at t13 and
x(t2)=0.08 at t23.
The sixth column shows the difference i.e.
∆x = x(t2) – x(t1) ;
The last column depicts the ratio of ∆x and ∆t, i.e. the average velocity with respect to the value of ∆t corresponding to the first column.
Table 1 depicting limiting value of ∆xt at t = 4 s
Table 1 shows that as we reduce the value of ∆t from 2.0 s to 0.010 s, the average velocity approaches the limiting value of 3.84 m/s which in turn is the value of velocity at dx t=4.0s, i.e.the value of dx/dt at t equals to 4.0s.
By this method, we can find out the velocity at each instant when the car is in motion as shown in Fig 2.
Fig 2 (a) Position versus time graph of a car for Uniform Motion
Fig 2 (a) Position versus time graph of a car for Non-Uniform Motion
The variation of velocity with time for this case is shown in Fig. 3.
Though the graphical method for determining the instantaneous velocity is not always a convenient method. In order to do this, we must carefully plot the position versus time graph and find out the value of the average velocity as ∆t value keeps on decreasing .
It becomes easier to calculate the value of velocity at different instants provided we have the data of all the positions at those instants. Otherwise it can also be expressed as functions of time on knowing the exact positions.
Henceforth we can find out the values of ∆x/∆t from the data when ∆t values are decreasing. We can calculate the limiting values as we have done previously in Table 1 or differential calculus can be used for the given expression and find out dx/dt at different instants.
For a clearer perspective looking at an example can be helpful.
Example 1.
Q)The position of an object moving along x-axis is given by x = a + bt₂ where a=8.5m,b=2.5ms–2 and is measured in seconds. What is its velocity at t = 0 s and t = 2.0 s. What is the average velocity between t = 2.0 s and t = 4.0 s ?
A) In notation of differential calculus, the velocity is
v=dx/dt =[ d{(a+bt₂)} /dt ]- 2bt =5.0t m/s
At t=0s, v=0m/s and
At t=2.0s, v=10m/s .
Average velocity
= {x (4.0) − x (2.0)}/ 4.0 − 2.0
={a+16b–a–4b} /2
= 6.0×b
=6.0×2.5=15m/s
▹ From Fig. 3, we know that during the period t =10 s to 18 s the velocity is constant. Between the period t =18 s to t = 20 s, it is uniformly decreasing and during the period t = 0 s to t = 10 s, it is increasing.
For uniform motion, velocity is equal to the average velocity at all instants.
Speed or instantaneous speed is the magnitude of velocity.
For example, a velocity of +22.0m/s and a velocity of (–22.0m/s) both have an associated speed of 22.0 m/s.
It is important to note that though the average speed over a finite interval of time can be greater than or equal to the average velocity. Therefore it can be said that the instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that particular instant.
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.
