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Instantaneous Speed

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You might be roaming out in a car. You take certain twists, turns then stop where signal are there changing your speed at each instant. At any moment or instant you desire to see what's the speed? Its called instantaneous speed. The speedometer tells you this.
Speedometer of Car
Speedometer showing the speed of the car at any instant of time in your trip.

Instantaneous Speed Definition

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The speed of a moving bodies varies at every moment but we just see whats its speed in average. But the speed may vary for one instant to the other. So instantaneous speed is defined as:
The rate at which the displacement takes place at the any desired instant of time.

The instantaneous speed of an object is not to be confused with the average speed. But can be calculated by finding average speed for short time interval or short distance.

Instantaneous Speed Formula

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Instantaneous speed formula is used to find the rate of change of displacement for any instant of time. It is just a first derivative with respect to time and is given as

Instantaneous Speed Formula

Where dx is the displacement and
dt is the time taken.
T is the that instant of time we are desiring the speed.
It is expressed as meter per second (m/s) or miles per hour.

Average Speed and Instantaneous Speed

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You may think that even though you can get to know about the speed using the average speed what's the use of instantaneous speed?
Lets take a boy trying for 100 m race. In his way he gets a stone and falls down later he recover from the pain within 5 seconds and starts running at some speed and at the end of his race he would be at his highest speed thus completes the race within 20 s. Here you could see at some point of time he was at rest but if we consider the average speed he would be at the speed of 5m/s even though he was having zero speed at some instant of time.

The average speed tells only about the total distance covered to the total time taken but while covering the total distance many a times it may happen that the moving body might be in rest at some point of time or may be at highest speed at some other instant of time. To calculate this we use the concept of Instantaneous Speed as it tells about the speed at any instant of time.

How to Find Instantaneous Speed?

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Instantaneous speed is calculated if the distance function x(t) and instant of time t is known using above formula. Here are given some instantaneous speed problems.

Solved Examples

Question 1: A particle undergoes the displacement given by the function x(t) = 6 t2 - 3t + 2. Find the instantaneous speed at time t = 5s.
Solution:
 
Given: The function is x(t) = 6 t2 - 3t + 2 and time t = 5 s
The instantaneous speed is given by
Instantaneous Speed = $\lim_{t \to 5}$ $\frac{(d(6 \times t^{2} - 3 \times t + 2)}{dt}$
                               = $\lim_{t \to 5}$  (6 $\times$ 2t - 3 + 0)
                               = 6 $\times$ 2 $\times$ 5 - 3
                               = 60 - 3
                               = 57 m/s
The Instantaneous speed of the particle is 57 m/s at 5th second.

 

Question 2: A body displaces at the rate of X = 4at2. Calculate the instantaneous speed of the body at t = 3 seconds if the a value is 5 m/s2 ?
Solution:
 
Given: Distance x = 4at2 and time t = 3 s
The instantaneous speed is given by
Instantaneous Speed = $\lim_{t \to 3}$ $\frac{d(4at^2)}{dt}$
                               = $\lim_{t \to 3}$ (4a $\times$ (2t))
                               = 4a $\times$ 2t
Putting the values of a and t we have
Instantaneous Speed = 4 $\times$ 5 $\times$ 2 $\times$ 3
                               = 120 m/s
The instantaneous speed of body at 3rd second is 120 m/s.
 

Examples of Instantaneous Speed

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Lets see some of the examples :

Two vehicles covers a distance of 100 m. Their average velocity will be the same but you could see how their speed varies for every time instant shown below graphically. Blue line represents the bike speed whereas pink line represents car speed.

Instantaneous Speed Graph
Vehicles
2s
4s
6s
8s
10s
12s
Car Speed (m/s)
5 18 12 12 215
Bike Speed (m/s)
10 28 20 14 85

The car is moving at some speed.The speedometer showing different readings at different intervals of time due to the change in cars speed due to the twist and turns in the path is as given in table.

Speed (m/s)
50
70
80
100
Time interval
5 8 10 12