How does Instantaneous Velocity differ from Average Velocity?
Last Updated : 21 Jun, 2024
Velocity is a crucial topic in physics. Many qualities of a body, such as kinetic energy and viscosity, are influenced by its velocity. The term velocity describes how quickly or slowly an object is moving. Velocity can be defined as the rate of change of the object’s position with respect to time and frame of reference.
In disciplines as diverse as kinematics, kinetics, dynamics, astrophysics, and engineering, the idea of velocity is usually applied. To excel in such disciplines, it is critical to have a thorough knowledge of the notions of instantaneous velocity and average velocity.
In this article, we will look closely at how Instantaneous Velocity differs from Average Velocity.
What is Instantaneous Velocity?
The rate of change of position over a relatively small interval of time is known as the instantaneous velocity or the velocity of an object at a certain time. The instantaneous velocity is arrived at by multiplying the object’s instantaneous speed by the direction in which it is moving at that particular time.
Velocity is defined as the rate of change of displacement in Classical physics. The terms “velocity” concern the vector quantity. They have magnitude which is a numerical value while the direction is also taken into consideration. An object with uniform velocity may thus be said to start with the same velocity as its standard velocity. For the instantaneous velocity, the International System of Units (SI) is meters per second (m/s).
It is determined as the arithmetic mean between the velocities and the minimum time required. Average velocity can be calculated using the parameter of the total displacement with regard to total time. The displacement directly depends on the time period taken for the displacement to occur. The actual rate of change of displacement with respect to time is called the instantaneous velocity and the Ratio of the limit of the change of displacement to the limit of the time interval is called instantaneous velocity.
V(t)=lim_{\bigtriangleup t\rightarrow 0}\frac{\bigtriangleup x}{\bigtriangleup t}=\frac{dx}{dt}
Where
- V(t) is Instantaneous velocity at time t
- x denotes displacement
- t denotes time
- Δt is the small-time interval
This formula implies that instantaneous velocity is the derivative of displacement with respect to time. It measures how quickly the position changes at a specific moment.
Graphical Interpretation
On displacement-time graph instantaneous velocity can be represented by the slope of the tangent to it at that particular point. The tangent line touches the curve only one point to give the rate of change of displacement at that point of time. This slope gives information both in the form of the rate at which the velocity is increasing or decreasing and the direction of this rate.
The slope of the tangent line to the function at that place is equal to the instantaneous velocity at any given point on a function x (t). The slope of the distance-time graph, often known as the x-t graph, can also be used to illustrate it. The instantaneous velocity formula is:
The velocity between two positions in a time limit where the time between them progressively becomes 0 is called instantaneous velocity. The location of x in relation to the function of t is denoted by X(t). The following is the equation for the average velocity between the two points:
V=\frac{[x(t_{2})-x(t_{1})]}{[(t_{2}-t_{1})]}
Let, t1 and t2 and t_{2} = t+\bigtriangleup t
The Δt must be 0 in order to determine instantaneous velocity,
Using \bigtriangleup t\rightarrow 0 as the limit and putting the expressions into the equation
V(t)=lim_{\bigtriangleup t\rightarrow 0}\frac{\bigtriangleup x}{\bigtriangleup t}=\frac{dx}{dt}
Thus, V(t) = dx/dt
Practical Application
Instantaneous velocity plays a vital role to comprehend and forecast the movement of bodies in real-life situations. For example, in Automotive engineering it is useful to know the instantaneous velocity of a car in designing anti-lock braking systems and traction control systems. In sports it helps the sporting personnel to enhance their performance especially in terms of speed at certain incidences.
What is Average Velocity?
Average velocity is defined by the total change division by the total time which is Delta Displacement/ Delta Time. Average velocity of an item is always lesser than or equal to average speed. This may be represented by pointing out that while distance will always be increasing, displacement can also be either increasing or decreasing or even zeroed at any place as required.
SI Unit of average velocity is meters per second The International System of Units (S. I) uses five fundamental units among which are the unit of , viz: time, mass, electrical current, thermodynamic temperature and length.
Average\space Velocity(v_{avg})=\frac{Total\space Displacement (\bigtriangleup x)}{Total \space Time \space Taken (\bigtriangleup t)}
v_{avg}=\frac{\bigtriangleup x}{\bigtriangleup t}= \frac{x_{1}-x_{0}}{t_{1}-t_{0}}
Where,
- x1 is the final displacement
- x0 is the initial displacement
- t1 is the time taken at final position
- t0 is the time taken at initial position
Graphical Interpretation
On a displacement-time graph, average velocity is shown as a straight line drawn through the points of initial displacement and the final displacement. This line is known as the secant line, as opposed to the tangent line which used for the instantaneous velocity.
Thus, when the displacement time graph is a straight line, then the average velocity will be equal to the instantaneous velocity. But in case of a curved displacement-time graph, it only provides a general idea about the velocity that has been achieved at any point of time as when displacement is changing, the velocity is not a fixed parameter.
The average velocity is defined as the slope of a secant line connecting two locations with t coordinates corresponding to the time period's borders.
The average velocity is the same as the velocity averaged over time, or its time-weighted average, which may be determined as the velocity's time integral.
v_{avg}=\frac{1}{t_{1}-t_{0}}\int_{t_{0}}^{t_{1}}v(t)dt
Where we can find out,
\bigtriangleup x=\int_{t_{0}}^{t_{1}}v(t)dt
And,
△t = t1 − t0
Practical Application
Average velocity is used in different real-life situations where the general movement is more crucial than the movement at a particular period. For instance, in the area of transportation, average velocity is used to make predictions regarding distance travel time. In physics experiments especially those concerning motion, it is easy to use and one does not need sophisticated timing equipment to perform it.
Difference between Instantaneous Velocity and Average Velocity
Feature | Instantaneous Velocity | Average Velocity |
|---|
Definition: | The rate of change of position over a relatively short time span is called instantaneous velocity. | The average velocity is calculated by dividing the change in total displacement by the total time taken |
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Change | It changes instantly with time | It remains the same as long as the total displacement remains same. |
|---|
Time interval | It corresponds to an infinitesimally small moment in time. | It often represents an interval of time, often denoted as Δt. |
|---|
Formula | V(t)=lim_{\bigtriangleup t\rightarrow 0}\frac{\bigtriangleup x}{\bigtriangleup t}=\frac{dx}{dt} | v_{avg}=\frac{\bigtriangleup x}{\bigtriangleup t}= \frac{x_{1}-x_{0}}{t_{1}-t_{0}} |
|---|
Physical Interpretation | It will indicate how fast and in what direction an object is moving at a given instant. | It represents the rate of change of displacement over time. |
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Application | Useful for analyzing dynamics in a specific instant | Useful for understanding the overall motion of an object during a period. |
|---|
Graphical interpretation | The slope of the distance-time graph, often known as the x-t graph, can also be used to illustrate instantaneous velocity. | The slope of the secant line connecting two points on the distance-time graph, or x-t graph, illustrates Average Velocity |
|---|
Examples | speedometer, speed trap, etc. | Cable car's average speed is actually average velocity stat, etc. |
|---|
Key Differences between Instantaneous Velocity and Average Velocity
Definition
- Instantaneous Velocity: The velocity at a specific instant.
- Average Velocity: The total displacement divided by the total time taken.
Calculation
- Instantaneous Velocity: Derived using the derivative of displacement with respect to time.
- Average Velocity: Calculated by dividing total displacement by total time.
Graphical Representation
- Instantaneous Velocity: Slope of the tangent to the displacement-time curve at a point.
- Average Velocity: Slope of the secant line connecting initial and final points on a displacement-time graph.
Dependence on Time Interval
- Instantaneous Velocity: Considers an infinitesimally small time interval.
- Average Velocity: Calculated over a finite time interval.
Conclusion
It is thus important for students in physics and engineering to be able to distinguish between instantaneous velocity and average velocity. Instantaneous velocity becomes helpful for the dynamic studies and for practical applications when giving the value of a velocity at a point of time in the motion of the body.
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Sample Questions
Question 1: When the position of the provided particle is x(t) = 2.0t + 0.73m, calculate the instantaneous velocity at t = 3.0s.
Solution:
Given
x(t) = 2.0t + 0.73m
t = 3 sec
\frac{dx}{dt}=\frac{d(2t+0.7t^{3})}{dt}
v = 2t + 3 × 0.7 × t2
v = 2t + 2.1t2
At t = 3sec
v = 2 × 3 + 3 × 0.7 × 32
v = 6 + 2.1 × 9
v = 6 + 18 - 9
v = 24.9 m/s
Question 2: Calculate the instantaneous velocity at the time (t) = 4 sec using the function x = 3t2 - 5t + 2 to get the position of a moving bus.
Solution:
Given,
x = 3t2 - 5t + 2
t = 4sec
V_{int}=\frac{dx}{dt}
V_{int}=\frac{d(3t^{2}-5t+2)}{dt}
V = 6t - 5
At t = 4 sec
V = 6t - 5
V = 6 × 4 - 5
V = 19 m/s
Question 3: What is the shortest time a predator will take to catch its prey at a distance of 200m if its average velocity is 90 km/hr?
Solution:
Given,
V_{tiger}= 90 km/hr
V = 90 × 5/18 m/sec
V = 25m/sec
a=\frac{v^{2}-u^{2}}{2s}
a=\frac{25^{2}-0^{2}}{2\times 200}
a = 625/400
a = 1.56 m/s
S=ut+\frac{1}{2}at^{2}
200=0\times t+\frac{1}{2}1.56\times t^{2}
t2 = 400/1.56
t2 = 256.4
t2 = 16 sec
Question 4: S(t) = 9t + 12t2 is the equation of motion for a car traveling in a straight path for 15 seconds before crashing. Calculate the instantaneous velocity at the 9th-second interval.
Solution:
Given,
S(t) = 9t + 12t2
\frac{ds}{dt}=\frac{d(9t+12t^{2})}{dt}
V = 9 + 24t
Therefore Vinst at t = 9
V = 9 + 24(9)
V = 225 m/s
Question 5: With a function, x = 7t2+ 3t + 3a given projectile moves in a straight line for time (t) = 2s. Calculate the instantaneous velocity of a moving object.
Solution:
Given,
x = 7t2 + 3t + 3
t = 2 sec
V_{inst}=\frac{dx}{dt}
V_{inst}=\frac{d(7t^{2}+3t+3)}{dt}
Vinst = 14t + 3
At t = 2 sec
V(2) = 14(2) + 3
V(2) = 31m/s
Question 6: In 4 minutes, the bus driver travels 15 kilometers down the road. He then took a step back and drove 9 kilometers down the road in 2 minutes. What is his typical speed?
Solution:
Average \space Velocity(v_{avg})=\frac{Total\space Displacement (\bigtriangleup x)}{Total \space Time \space Taken (\bigtriangleup t)}
Average Velocity (vavg) = \frac{(15-9)}{(4+2)}
Average Velocity (vavg) = 6/6
Average Velocity (vavg) =1 km/m
Question 7: Calculate a person's average velocity at a specific time when he walks 5 meters in 3 seconds and 15 meters in 5 seconds in a straight line along the x-axis.
Solution:
Given,
The initial distance traveled xi = 5 m,
Final distance traveled, xf = 15m,
Initial time interval = 3 s,
Final time interval tf = 5 s,
v_{avg}=\frac{\bigtriangleup x}{\bigtriangleup t}= \frac{x_{1}-x_{0}}{t_{1}-t_{0}}
v_{avg}=\frac{\bigtriangleup x}{\bigtriangleup t}= \frac{(15-5)}{(5-3)}
Average Velocity (vavg) = 5 m/s.