How to find instantaneous velocity from average velocity: Detailed Insights

How to Find Instantaneous Velocity from Average Velocity

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Velocity is an important concept in physics that measures how fast an object is moving and in which direction. It plays a crucial role in understanding the motion of objects and calculating various other quantities such as acceleration, displacement, and momentum. In this blog post, we will explore the relationship between average velocity and instantaneous velocity, and learn how to find instantaneous velocity from average velocity.

How to Calculate Average Velocity

Average velocity is defined as the displacement of an object divided by the time it takes for that displacement to occur. It gives us a measure of the overall change in position of an object over a specific time interval. The formula for average velocity is:

$v_{avg} = frac{Delta x}{Delta t}$

where $v_{avg}$ represents the average velocity, $Delta x$ represents the change in position, and $Delta t$ represents the change in time.

To calculate average velocity, simply subtract the initial position from the final position and divide it by the time interval. For example, if an object travels a distance of 100 meters in a time of 20 seconds, the average velocity can be calculated as:

$v_{avg} = frac{100 , text{m}}{20 , text{s}} = 5 , text{m/s}$

How to Determine Instantaneous Velocity

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While average velocity gives us an overall measure of an object’s motion over a specific time interval, instantaneous velocity provides us with information about the object’s motion at a specific point in time. It is the velocity of an object at an exact moment or instant. To determine instantaneous velocity, we need to consider the concept of limit.

The Concept of Limit in Calculating Instantaneous Velocity

The concept of limit involves finding the value that a function approaches as the input approaches a certain value. In the context of instantaneous velocity, we want to find the velocity of an object at a particular moment, which corresponds to an infinitesimally small time interval. By taking smaller and smaller time intervals, we can approximate the instantaneous velocity.

The Formula for Instantaneous Velocity

Instantaneous velocity can be calculated using calculus, specifically by taking the derivative of the position function with respect to time. In mathematical terms, the formula for instantaneous velocity is:

$v(t) = lim_{Delta t to 0} frac{Delta x}{Delta t}$

where $v(t)$ represents the instantaneous velocity at time $t$ , $Delta x$ represents the change in position, and $Delta t$ represents the change in time. Taking the limit as $Delta t$ approaches zero allows us to calculate the velocity at an exact moment.

Worked out Example on Instantaneous Velocity Calculation

Let’s consider an example to better understand how to find instantaneous velocity from average velocity. Suppose an object is moving along a straight line, and its position at time $t$ is given by the equation $x(t) = 3t^2 + 2t + 1$ . We want to find the instantaneous velocity at $t = 2$ seconds.

To find the instantaneous velocity, we need to take the derivative of the position function with respect to time:

$v(t) = frac{d}{dt} (3t^2 + 2t + 1)$

Using the power rule of differentiation, we can differentiate each term of the equation:

$v(t) = 6t + 2$

Now, we can substitute $t = 2$ into the equation to find the instantaneous velocity at that moment:

$v(2) = 6(2) + 2 = 14 , text{m/s}$

Therefore, the instantaneous velocity at $t = 2$ seconds is $14 , text{m/s}$ .

Comparing Instantaneous Velocity and Average Velocity

Instantaneous velocity and average velocity are related but represent different aspects of an object’s motion.

Situations when Instantaneous Velocity equals Average Velocity

In certain cases, the instantaneous velocity and average velocity of an object can be equal. This occurs when the object is moving at a constant velocity over a specific time interval. For example, if a car travels at a constant speed of $50 , text{km/h}$ for $2 , text{hours}$ , the average velocity over the entire time interval would be $50 , text{km/h}$ . Since the car maintains a constant velocity, the instantaneous velocity at any point during the $2 , text{hours}$ would also be $50 , text{km/h}$ .

Practical Examples Illustrating the Comparison

To further illustrate the difference between instantaneous velocity and average velocity, let’s consider a couple of practical examples.

Example 1: A car travels at a speed of $60 , text{km/h}$ for $1 , text{hour}$ . During the first $30 , text{minutes}$ , it maintains a constant velocity of $60 , text{km/h}$ . However, during the second $30 , text{minutes}$ , it comes to a complete stop and remains stationary. In this case, the average velocity over the entire $1 , text{hour}$ would still be $60 , text{km/h}$ , but the instantaneous velocity during the first $30 , text{minutes}$ would be $60 , text{km/h}$ and during the second $30 , text{minutes}$ would be $0 , text{km/h}$ .

Example 2: A ball is thrown upwards with an initial velocity of $20 , text{m/s}$ . As it rises, its velocity gradually decreases until it reaches its peak height and starts falling back down. At the highest point, the ball momentarily comes to a stop before accelerating downwards. In this case, the average velocity over the entire trajectory would be $0 , text{m/s}$ since the ball starts and ends at the same height with the same velocity. However, the instantaneous velocity would be positive during the ascent, $0 , text{m/s}$ at the highest point, and negative during the descent.

Understanding the relationship between average velocity and instantaneous velocity is crucial for analyzing the motion of objects. By calculating average velocity, we can determine the overall change in position over a specific time interval. To find instantaneous velocity, we need to consider the concept of limit and take the derivative of the position function with respect to time. It is important to remember that instantaneous velocity provides information about an object’s motion at an exact moment, while average velocity gives an overall measure of its motion over a specific interval.

What is the difference between instantaneous velocity and average velocity? Provide insight into the content of the article Difference between instantaneous velocity and average velocity.

The difference between instantaneous velocity and average velocity is explored in detail in the article Difference between instantaneous velocity and average velocity. Instantaneous velocity refers to the velocity of an object at a specific moment in time, while average velocity is the total displacement of an object divided by the total time taken. The article further delves into the calculation methods, applications, and significance of these two concepts in physics. By understanding the difference between instantaneous velocity and average velocity, one can gain a comprehensive understanding of how velocity is measured and analyzed.

Numerical Problems on how to find instantaneous velocity from average velocity

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Problem 1:

A car travels along a straight road for 4 hours and covers a distance of 320 km. Find the instantaneous velocity at the end of the second hour if the average velocity during the first 3 hours is 80 km/hr.

Solution:

Let’s assume that the instantaneous velocity at the end of the second hour is $v_2$ km/hr.

The average velocity during the first 3 hours is given by:

$v_{text{avg}} = frac{text{Total distance}}{text{Total time}}$

Substituting the given values, we have:

$80 , text{km/hr} = frac{320 , text{km}}{3 , text{hours}}$

Solving the above equation, we find that the total distance covered in the first 3 hours is 240 km.

To find the instantaneous velocity at the end of the second hour, we need to determine the distance covered in the second hour. Since the car has covered a total distance of 240 km in the first 3 hours, the distance covered in the second hour is given by:

$240 , text{km} - 80 , text{km/hr} times 2 , text{hours} = 80 , text{km}$

Therefore, the instantaneous velocity at the end of the second hour is 80 km/hr.

Problem 2:

A cyclist covers a distance of 150 km in 5 hours. Find the instantaneous velocity after 3 hours if the average velocity during the first 4 hours is 40 km/hr.

Solution:

Let’s assume that the instantaneous velocity after 3 hours is $v_3$ km/hr.

The average velocity during the first 4 hours is given by:

$v_{text{avg}} = frac{text{Total distance}}{text{Total time}}$

Substituting the given values, we have:

$40 , text{km/hr} = frac{150 , text{km}}{4 , text{hours}}$

Solving the above equation, we find that the total distance covered in the first 4 hours is 160 km.

To find the instantaneous velocity after 3 hours, we need to determine the distance covered in the third hour. Since the cyclist has covered a total distance of 160 km in the first 4 hours, the distance covered in the third hour is given by:

$160 , text{km} - 40 , text{km/hr} times 3 , text{hours} = 40 , text{km}$

Therefore, the instantaneous velocity after 3 hours is 40 km/hr.

Problem 3:

A train covers a distance of 600 km in 8 hours. Find the instantaneous velocity after 6 hours if the average velocity during the first 5 hours is 70 km/hr.

Solution:

Let’s assume that the instantaneous velocity after 6 hours is $v_6$ km/hr.

The average velocity during the first 5 hours is given by:

$v_{text{avg}} = frac{text{Total distance}}{text{Total time}}$

Substituting the given values, we have:

$70 , text{km/hr} = frac{600 , text{km}}{5 , text{hours}}$

Solving the above equation, we find that the total distance covered in the first 5 hours is 350 km.

To find the instantaneous velocity after 6 hours, we need to determine the distance covered in the sixth hour. Since the train has covered a total distance of 350 km in the first 5 hours, the distance covered in the sixth hour is given by:

$350 , text{km} - 70 , text{km/hr} times 5 , text{hours} = 50 , text{km}$

Therefore, the instantaneous velocity after 6 hours is 50 km/hr.

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Raghavi Acharya

I am Raghavi Acharya, I have completed my post-graduation in physics with a specialization in the field of condensed matter physics. I have always considered Physics to be a captivating area of study and I enjoy exploring the various fields of this subject. In my free time, I engage myself in digital art. My articles are aimed towards delivering the concepts of physics in a very simplified manner to the readers.