Average speed and instantaneous velocity

Speed and velocity are two different entities. Average speed is the total path length traversed by an object divided by the time taken for the motion. Instantaneous speed is the speed recorded at a specific point in time. 

On the other hand, average velocity implies the change in position of an object (displacement) divided by the time the displacement has taken place. So, we can define instantaneous velocity 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.

Now let’s delve into the average speed and instantaneous velocity study material.

Average speed

The average speed is defined as the distance covered by the object divided by the amount of time elapsed since it started. 

Since it is defined solely by a magnitude, the average speed is unquestionably a scalar quantity.

The average speed is calculated using the following formula:

Average speed = Total distance covered throughout the journeyTotal time is taken for the journey

Miles per hour (mph), kilometres per hour (km/h), metres per second (m/s) and feet per second (ft/s) are some of the most often used units of speed.

The average speed of your brand-new red sports automobile, according to your friend’s assessment, was perfect! He divided the distance travelled by car (45 miles) by the time elapsed (1.25 hours). The work on the highway, as well as a slew of red lights on the side roads, significantly hampered your progress. As a result of the long elapsed period, the average speed was low. Mistakes and assumptions that are often made while calculating the same are defined in the next section.

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. 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

vinstant = x(t2) – x(t1)t2 – t1 

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.

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 Average speed and instantaneous velocity

Q1: A car moves at a speed of 30 km/hr for 2 hours and then slows down to 20 km/hr for the next 1 hour. Find the average speed of the car.

Solution: Distance 1 = 30x 2 = 60 km 

Distance 2 = 20x 1 =20 km

Distance total = Distance 1 + Distance 2 

D = 60 + 20 = 80 km

Total distance travelled / Total time taken = Average speed = 80/3 = 26.67 km/hr

Q2: 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

Conclusion

Study material notes on average speed and instantaneous velocity, i.e. they are two distinct quantity. The average speed doesn’t need to have the same magnitude as the instantaneous velocity. However, Average speed relies on distance, and instantaneous velocity depends on displacement. 

Average speed and instantaneous velocity are not the same things. 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 that can be used by physicists, mathematicians, statisticians to solve problems related to our day to day lives and understand the physics behind it.