You are viewing the article How to Calculate Instantaneous Velocity at Thptlaihoa.edu.vn you can quickly access the necessary information in the table of contents of the article below.
When studying the motion of objects, it is often important to understand the concept of velocity. Velocity is defined as the rate at which an object changes its position in a specific direction. While average velocity provides us with an overall understanding of an object’s motion over a certain time interval, instantaneous velocity allows us to analyze its motion at any given moment. In this guide, we will delve into the topic of calculating instantaneous velocity, exploring the mathematical methods and concepts required to accurately determine an object’s velocity at a specific point in time. Understanding how to calculate instantaneous velocity is essential for various fields, such as physics, engineering, and even everyday observations of moving objects. So, let us embark on this journey to unravel the mystery behind instantaneous velocity!
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 25 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 92,696 times.
Velocity is defined as the speed of an object in a specified direction. [1] XResearch Source In many cases, to find the velocity we will use the equation v = s/t, where v is the velocity, s is the total distance traveled by the object from the initial position, and t is the time it takes the object to travel that distance. However, in theory this formula only gives the average velocity of the object over the distance. By calculus we can calculate the speed of the object at any time along the distance. That’s the instantaneous velocity and is defined by the equation v = (ds)/(dt) , or in other words, it’s the derivative of the equation for the average velocity. [2] XResearch Source
Steps
Calculate instantaneous velocity
![Image titled Calculate Instantaneous Velocity Step 1](https://www.wikihow.com/images/thumb/1/1f/Calculate-Instantaneous-Velocity-Step-1-Version-2.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-1-Version-2.jpg)
s = -1.5t 2 + 10t + 4
- In this equation, the variables are:
-
- s = displacement distance . Distance of the moving object from the starting position. For example, if an object travels 10 meters forward and 7 meters backward, its total distance traveled is 10 – 7 = 3 meters (not 10 + 7 = 17m).
- t = time . This variable simply needs no explanation, usually measured in seconds.
-
![Image titled Calculate Instantaneous Velocity Step 2](https://www.wikihow.com/images/thumb/d/d1/Calculate-Instantaneous-Velocity-Step-2-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-2-Version-4.jpg)
- In other words, start to differentiate from left to right on the “t” side of the equation. Every time you encounter the variable “t”, you subtract the exponent by 1 and multiply the whole term by the original exponent. Any constant terms (terms without “t”) will disappear because they are multiplied by 0. The process is actually not as difficult as you might think – let’s take the equation in the step above as an example:
s = -1.5t 2 + 10t + 4
(2)-1.5t (2-1) + (1)10t 1 – 1 + (0)4t 0
-3t 1 + 10t 0
-3t + 10
![Image titled Calculate Instantaneous Velocity Step 3](https://www.wikihow.com/images/thumb/b/b4/Calculate-Instantaneous-Velocity-Step-3-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-3-Version-4.jpg)
- In the example above, the derivative of the equation would look like this:
ds/dt = -3t + 10
![Image titled Calculate Instantaneous Velocity Step 4](https://www.wikihow.com/images/thumb/0/06/Calculate-Instantaneous-Velocity-Step-4-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-4-Version-4.jpg)
ds/dt = -3t + 10
ds/dt = -3(5) + 10
ds/dt = -15 + 10 = -5 meters/second
- Note that we use the “meter/second” unit above. Since we are solving a problem with displacement in meters and time in seconds, where velocity is the distance traveled in time, this unit is appropriate.
Graph instantaneous velocity estimate
![Image titled Calculate Instantaneous Velocity Step 5](https://www.wikihow.com/images/thumb/8/86/Calculate-Instantaneous-Velocity-Step-5-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-5-Version-4.jpg)
- To graph the distance traveled, you use the x-axis as the time and the y-axis as the distance traveled. You then determine a number of points by substituting the values of t into the equation of motion, the resulting s values, and you plot the points t,s (x,y) on the graph.
- Note that the graph can extend below the x-axis. If the line of motion of an object goes below the x-axis, this means that the object is moving backwards from its original position. In general, graphs won’t extend behind the y-axis – we don’t normally measure the velocity of an object moving backwards over time!
![Image titled Calculate Instantaneous Velocity Step 6](https://www.wikihow.com/images/thumb/e/ea/Calculate-Instantaneous-Velocity-Step-6-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-6-Version-4.jpg)
- Assume the distance traveled has points (1,3) and (4,7). In this case, if we want to find the slope at (1,3) we can set (1;3) = P and (4;7) = Q .
![Image titled Calculate Instantaneous Velocity Step 7](https://www.wikihow.com/images/thumb/0/02/Calculate-Instantaneous-Velocity-Step-7-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-7-Version-4.jpg)
H = (y Q – y P )/(x Q – x P )
H = (7 – 3)/(4 – 1)
H = (4)/(3) = 1.33
![Image titled Calculate Instantaneous Velocity Step 8](https://www.wikihow.com/images/thumb/b/b8/Calculate-Instantaneous-Velocity-Step-8-Version-2.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-8-Version-2.jpg)
Q = (2,4,8): H = (4.8 – 3)/(2 – 1)
H = (1,8)/(1) = 1.8Q = (1.5;3.95): H = (3.95 – 3)/(1.5 – 1)
H = (0.95)/(0.5) = 1.9Q = (1,25;3.49): H = (3.49 – 3)/(1.25 – 1)
H = (0.49)/(0.25) = 1.96
![Image titled Calculate Instantaneous Velocity Step 9](https://www.wikihow.com/images/thumb/f/f5/Calculate-Instantaneous-Velocity-Step-9-Version-2.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-9-Version-2.jpg)
- In the above example, as we move H closer to P, we have H values of 1.8; 1.9 and 1.96. Since these numbers are approaching 2, we can say 2 is the approximate value of the slope at P.
- Remember that the slope at any point on the graph is the derivative of the equation of the graph at that point. Since the graph represents the distance traveled by an object over time, as we saw in the previous section, its instantaneous velocity at any given point is the derivative of the distance the object has moved at the point in question. approach, we can say that 2 meters/second is an approximate estimate of the instantaneous velocity when t = 1.
Sample problem
![Image titled Calculate Instantaneous Velocity Step 10](https://www.wikihow.com/images/thumb/7/7e/Calculate-Instantaneous-Velocity-Step-10-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-10-Version-4.jpg)
- First, take the derivative of the equation:
s = 5t 3 – 3t 2 + 2t + 9
s = (3)5t (3 – 1) – (2)3t (2 – 1) + (1)2t (1 – 1) + (0)9t 0 – 1
15t (2) – 6t (1) + 2t (0)
15t (2) – 6t + 2 - Then we substitute the value of t (4) in:
s = 15t (2) – 6t + 2
15(4) (2) – 6(4) + 2
15(16) – 6(4) + 2
240 – 24 + 2 = 22 meters/second
![Image titled Calculate Instantaneous Velocity Step 11](https://www.wikihow.com/images/thumb/5/5a/Calculate-Instantaneous-Velocity-Step-11-Version-4.jpg/v4-728px-Calculate-Instantaneous-Velocity-Step-11-Version-4.jpg)
- First, we find the points Q when t = 2; 1.5; 1.1 and 1.01.
s = 4t 2 – t
t = 2: s = 4(2) 2 – (2)
4(4) – 2 = 16 – 2 = 14, so Q = (2,14)t = 1.5: s = 4(1.5) 2 – (1.5)
4(2.25) – 1.5 = 9 – 1.5 = 7.5, so Q = (1.5;7.5)t = 1,1: s = 4(1,1) 2 – (1,1)
4(1.21) – 1.1 = 4.84 – 1.1 = 3.74, so Q = (1,1;3.74)t = 1.01: s = 4(1.01) 2 – (1.01)
4(1.0201) – 1.01 = 4.0804 – 1.01 = 3.0704, so Q = (1.01;3.0704) - Next we will get the H values:
Q = (2,14): H = (14 – 3)/(2 – 1)
H = (11)/(1) = 11Q = (1.5;7.5): H = (7.5 – 3)/(1.5 – 1)
H = (4,5)/(0.5) = 9Q = (1,1;3.74): H = (3.74 – 3)/(1,1 – 1)
H = (0.74)/(0.1) = 7.3Q = (1.01;3.0704): H = (3.0704 – 3)/(1.01 – 1)
H = (0.0704)/(0.01) = 7.04 - Since the H values seem to approach 7, we can say that 7 m/s is an approximate estimate of the instantaneous velocity at coordinates (1;3).
Advice
- To find the acceleration (change in velocity with respect to time), use the method in part one to take the derivative of the displacement equation. Then take the derivative again for the derivative equation just found. As a result, you have an equation that finds the acceleration at a given time – all you have to do is plug in the time value.
- The equation representing the correlation between Y (displacement) and X (time) can be very simple, like Y = 6x + 3. In this case, the slope is constant and it is not necessary to take it. derivative to calculate the slope, that is, it follows the form of the basic equation Y = mx + b for a graph of a linear line, ie the slope is 6.
- Distance traveled is like distance but has direction, so it is a vector quantity, and speed is a scalar quantity. Distance traveled can be negative, while distance is only positive.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 25 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 92,696 times.
Velocity is defined as the speed of an object in a specified direction. [1] XResearch Source In many cases, to find the velocity we will use the equation v = s/t, where v is the velocity, s is the total distance traveled by the object from the initial position, and t is the time it takes the object to travel that distance. However, in theory this formula only gives the average velocity of the object over the distance. By calculus we can calculate the speed of the object at any time along the distance. That’s the instantaneous velocity and is defined by the equation v = (ds)/(dt) , or in other words, it’s the derivative of the equation for the average velocity. [2] XResearch Source
In conclusion, calculating instantaneous velocity is a fundamental concept in physics that helps us understand the motion of objects at any given moment. By using calculus and the derivative of the position-time function, we can determine the rate of change of an object’s position and calculate its instantaneous velocity. This concept is particularly useful in scenarios where the speed or direction of an object changes continuously. Whether it is the motion of vehicles on the road or the trajectory of a projectile, understanding instantaneous velocity allows us to accurately describe and predict the behavior of moving objects. With the application of calculus and a thorough understanding of the concept, we can calculate instantaneous velocity with precision, contributing to our knowledge of the physical world around us.
Thank you for reading this post How to Calculate Instantaneous Velocity at Thptlaihoa.edu.vn You can comment, see more related articles below and hope to help you with interesting information.
Related Search:
1. What is instantaneous velocity?
2. Formula for calculating instantaneous velocity
3. Examples of calculating instantaneous velocity
4. How does instantaneous velocity differ from average velocity?
5. How to find instantaneous velocity from a position-time graph
6. How to calculate instantaneous velocity from a velocity-time graph
7. How to calculate instantaneous velocity from a distance-time graph
8. How to calculate instantaneous velocity using calculus
9. What factors can affect the calculation of instantaneous velocity?
10. Real-life applications of calculating instantaneous velocity.