# Instantaneous Velocity Calculator

The Instantaneous Velocity Calculator is a free online tool used to find the instantaneous velocity for a given displacement and time. STUDYQUERIES’s online instantaneous velocity calculator tool makes calculation faster, and it displays instantaneous velocity in a fraction of a second.

## How to Use the Instantaneous Velocity Calculator?

To use the instantaneous velocity calculator, follow these steps:

• Step 1: Enter the displacement, time, and t into the appropriate input fields
• Step 2: Click “Calculate Instantaneous Velocity” to get the result
• Step 3: The instantaneous velocity will be displayed in the output field

## What Is Instantaneous Velocity?

The instantaneous velocity, or simply velocity, is the quantity that tells us how fast an object is moving along its path. The average velocity between two points on the path in the limit where the time (and therefore the displacement) between two points approaches zero. We can illustrate the idea mathematically by expressing position $$x$$ as a continuous function of t denoted by $$x(t)$$. We can find the average velocity between two points by using this notation

$$\vec{v}=\frac{x(t_2)-x(t_1)}{t_2-t_1}$$

To find the instantaneous velocity at any position, we let $$t_1=t$$ and $$t_2=t+\Delta{t}$$. After inserting these expressions into the equation for the average velocity and taking the limit as $$\Delta{t}\longrightarrow 0$$, we find the expression for the instantaneous velocity:

$$v(t)=\lim_{\Delta{t} \to 0}\frac{x(t+\Delta{t})-x(t)}{\Delta{t}}=\frac{dx(t)}{dt}$$

The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to $$t$$:

$$v(t)=\frac{dx(t)}{dt}$$

Like average velocity, the instantaneous velocity is a vector with a dimension of length per time. The instantaneous velocity at a specific time point $$t_0$$ is the rate of change of the position function, which is the slope of the position function $$x(t)$$ at $$t_0$$. Below the Figure shows how the average velocity $$\vec{v}=\frac{\Delta x}{\Delta t}$$ between two times approaches the instantaneous velocity at $$t_0$$.

Unit Circle Calculator

Transformation Calculator

Integration By Parts Calculator

The instantaneous velocity is shown at the time $$t_0$$, which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times, $$t_1, t_2$$, and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative.

If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.

The graph shows position plotted versus time. Position increases from $$t_1\ to\ t_2$$ and reaches a maximum at $$t_0$$. It decreases to at and continues to decrease at $$t_4$$. The slope of the tangent line at t0 is indicated as the instantaneous velocity.

In the above figure graph of position versus time, the instantaneous velocity is the slope of the tangent line at a given point. The average velocities

$$\vec{v}=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}$$

between times $$\Delta{t}=t_6-t_1$$,$$\Delta{t}=t_5-t_2$$, and $$\Delta{t}=t_4-t_3$$ are shown. When $$\Delta t\longrightarrow 0$$, the average velocity approaches the instantaneous velocity at $$t_0$$.

### Instantaneous Velocity: Finding Velocity From A Position-Time Graph

Given the position-time graph, find the velocity-versus-time graph.

As a function of time, the position at minutes is plotted as a function of kilometers. Starting at the origin, it reaches $$0.5$$ kilometers at $$0.5$$ minutes, remains constant between $$0.5$$ and $$0.9$$ minutes, and decreases to $$0$$ at $$2.0$$ minutes.

As the object moves in the positive direction, it stops for a short while, then reverses direction, heading back toward the origin.  The object comes to rest instantly, which would require an infinite force. As a result, the graph approximates motion in the real world.

The graph comprises three straight lines. It displays them during three different time intervals. By taking the slope of the line using the grid, we can calculate the velocity during each time interval.

Solution:

Time interval 0 s to 0.5 s: $$\vec{v}=\frac{\Delta x}{\Delta t}$$

$$=\frac{0.5m−0.0m}{0.5s−0.0s}$$

$$=1.0m/s$$

Time interval 0.5 s to 1.0 s: $$\vec{v}=\frac{\Delta x}{\Delta t}$$

$$=\frac{0.0m−0.0m}{0.5s−1.0s}$$

$$=0.0m/s$$

Time interval 1.0 s to 2.0 s: $$\vec{v}=\frac{\Delta x}{\Delta t}$$

$$=\frac{0.0m−0.5m}{2.0s−1.0s}$$

$$=-0.5m/s$$

The graph of these values of velocity versus time is shown in the figure below:

Significance: During the time interval between $$0-sec$$ and $$0.5-sec$$, the object’s position is moving away from the origin, and the position-versus-time curve is positive. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is $$+1 \frac{m}{s}$$, as shown in the figure.

In the subsequent time interval, between $$0.5-sec$$ and $$1.0-sec$$, the position doesn’t change and we see the slope is zero. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is $$−0.5 \frac{m}{s}$$. It has reversed direction and a negative velocity.

### Speed

Most people use the terms speed and velocity interchangeably in everyday speech. However, in physics, they are distinct and have different meanings. Speed is a scalar; that is, it has no direction.

The average speed can be calculated by dividing the total distance traveled by the elapsed time:

$$Average\ speed=V_{avg}=\frac{Total\ distance}{Elapsed\ time}$$

The average speed is not necessarily the same as the average velocity, which is determined by dividing the total displacement by the elapsed time.

The average velocity for a trip that starts and ends at the same place is zero because the total displacement is zero. Due to the fact that the total distance traveled is greater than zero, the average speed is not zero. For example, if we are driving 300 km and must arrive at our destination by a certain time, we would be interested in our average speed.

We can calculate the instantaneous speed, however, from the magnitude of the instantaneous velocity:

$$Instantaneous\ speed=\vert{v(t)}\vert$$

If a particle is moving along the x-axis at $$+7.0 \frac{m}{s}$$ and another particle is moving along the same axis at $$-7.0 \frac{m}{s}$$, they have different velocities, but both have the same speed of $$7.0 \frac{m}{s}$$.

### Instantaneous Velocity Versus Speed

Consider the motion of a particle in which the position is $$x(t)=(3.0t−3t^2)m$$, then

• What is the instantaneous velocity at $$t = 0.25 s$$, $$t = 0.50 s$$, and $$t = 1.0 s$$?
• What is the speed of the particle at these times?

Strategy: The instantaneous velocity is the derivative of the position function, and the speed is its magnitude. Solving for instantaneous velocity is shown in the figure above.

Solution:

1. $$v(t)=\frac{dx(t)}{dt}=(3.0−6t)\frac{m}{sec}$$

$$v(0.25 s)=(3.0−6\times 0.25)\frac{m}{sec}=1.50\frac{m}{sec}$$

$$v(0.50 s)=(3.0−6\times 0.50)\frac{m}{sec}=0.00\frac{m}{sec}$$

$$v(1.0 s)=(3.0−6\times 1.0)\frac{m}{sec}=-3.0\frac{m}{sec}$$

2. $$Speed=\vert{v(t)}\vert=1.50\frac{m}{sec}, 0.0\frac{m}{sec},\ and\ 3.0\frac{m}{sec}$$

Significance

Using the particle’s velocity, we can determine whether it is moving to the left (west) or right (east). Speed indicates the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can visualize these concepts. The particle in (a) moves in the positive direction until $$t = 0.5 s$$ when it reverses direction.

The reversal of direction can also be seen in (b) at $$0.5 s$$ where the velocity is zero and then turns negative. At $$1.0 s$$ it is back at the origin where it started. The particle’s velocity at $$1.0 s$$ in (b) is negative because it is traveling in the negative direction. In (c), however, the speed is positive and remains positive for the entire journey. Velocity can also be interpreted as the slope of the position-versus-time graph.

The slope of $$x(t)$$ is decreasing toward zero, becoming zero at $$0.5 s$$ and increasingly negative thereafter. The analysis of comparing the position, velocity, and speed graphs helps detect errors in calculations. It is important that the graphs match each other so that calculations can be interpreted.

READ ALSO: 7 Useful Apps for College Students

## Instantaneous Velocity Calculator (Calculus):

An instantaneous velocity calculator in calculus is a tool that helps determine the velocity of an object at a specific instant in time. It utilizes calculus concepts, such as derivatives, to calculate the instantaneous velocity. By providing the necessary input, such as position-time data or a function representing the object’s motion, the calculator performs the required calculations to find the instantaneous velocity at a given point.

Example: Suppose you have the position-time function of an object given by f(t) = 2t^3 – 3t^2 + 5t. To find the instantaneous velocity at t = 2 seconds, you can use an instantaneous velocity calculator in calculus. By taking the derivative of the position function with respect to time (df(t)/dt), the calculator determines the derivative function, which represents the velocity function. Evaluating the velocity function at t = 2 seconds will provide the instantaneous velocity at that point.

## Instantaneous Velocity Formula Calculator:

An instantaneous velocity formula calculator is a tool that calculates the velocity of an object at a specific instant using a given formula. It is particularly useful when you have a formula that directly relates the object’s velocity to time or other variables. By plugging in the relevant values into the formula, the calculator provides the instantaneous velocity result.

Example: Consider the formula for the velocity of an object given by v(t) = 4t – 2. To calculate the instantaneous velocity at t = 3 seconds, you can use an instantaneous velocity formula calculator. By substituting t = 3 into the velocity formula, the calculator will evaluate the expression and provide the instantaneous velocity as 10 m/s.

## Instantaneous Velocity at a Point Calculator:

An instantaneous velocity at a point calculator is a tool that determines the velocity of an object precisely at a particular point in time. It is specifically designed to calculate the velocity at a given instant without requiring a function or formula. By inputting the necessary information, such as initial position, time, and acceleration, the calculator performs the calculations and provides the instantaneous velocity at the specified point.

Example: Suppose an object starts from rest, and after 5 seconds of acceleration at a rate of 3 m/s^2, you want to find the instantaneous velocity at that time using an instantaneous velocity at a point calculator. By inputting the initial position as 0, the time as 5 seconds, and the acceleration as 3 m/s^2, the calculator will compute the velocity at that instant and provide the result as 15 m/s.

## Instantaneous Velocity Formula from Graph:

An instantaneous velocity formula from a graph calculator is a tool that derives the velocity function based on a graph of the object’s position or displacement versus time. It analyzes the slope of the graph at various points to determine the velocity function or instantaneous velocities at specific instances.

Example: Consider a position-time graph showing the motion of an object. By using an instantaneous velocity formula from a graph calculator, you can determine the velocity at different points. The calculator analyzes the slope of the graph at each point to calculate the instantaneous velocity. For example, if the slope at t = 2 seconds is determined to be 5 m/s, the calculator will provide that as the instantaneous velocity at that point.

## Find the Instantaneous Velocity Calculator:

A “find the instantaneous velocity” calculator is a tool that assists in determining the velocity of an object at a specific instant in time. It helps calculate the instantaneous velocity by utilizing various methods, such as calculus or position-time data analysis. This type of calculator allows you to find the instantaneous velocity by providing the required input based on the available information or function.

Example: Suppose you have the position-time data for an object’s motion. By using a “find the instantaneous velocity

” calculator, you can input the position-time data points and select the desired time to find the instantaneous velocity. The calculator will perform the necessary calculations, such as differentiating the position function or analyzing the data points, to provide the instantaneous velocity at the specified time.

## Instantaneous Velocity Limit Calculator:

An instantaneous velocity limit calculator is a tool that calculates the velocity of an object at a particular instant by taking the limit as the time interval approaches zero. This concept is derived from calculus and involves finding the derivative of the position function to determine the instantaneous velocity.

Example: Suppose you have the position function of an object given by p(t) = t^2 + 3t – 2. By using an instantaneous velocity limit calculator, you can find the instantaneous velocity at a specific time, such as t = 2 seconds. The calculator will calculate the limit of the average velocity as the time interval approaches zero to obtain the instantaneous velocity at that point.

## Instantaneous Velocity Formula Calculus:

The instantaneous velocity formula in calculus refers to the mathematical expression used to calculate the velocity of an object at a specific instant. It involves taking the derivative of the object’s position function with respect to time to obtain the instantaneous velocity function.

Example: Consider the position function of an object given by s(t) = 5t^2 – 4t + 2. In calculus, to find the instantaneous velocity, you differentiate the position function with respect to time (ds(t)/dt). Applying the power rule, the derivative yields the instantaneous velocity function v(t) = 10t – 4.

## Instantaneous Velocity Example:

An instantaneous velocity example refers to a specific scenario or problem where you need to calculate the velocity of an object at a precise moment in time. It typically involves using calculus concepts, such as differentiation, or specific formulas to find the instantaneous velocity.

Example: Suppose a car is moving along a straight line, and its position function is given by p(t) = 3t^2 + 2t + 1. To find the instantaneous velocity at t = 2 seconds, you can use calculus or an instantaneous velocity calculator. By differentiating the position function with respect to time, you obtain the instantaneous velocity function v(t) = 6t + 2. Evaluating v(2) will give the instantaneous velocity at t = 2 seconds, which is 14 m/s.

## Instantaneous Calculator:

An instantaneous calculator, in a general sense, refers to any calculator or computational tool that helps perform instantaneous calculations. It could be a specific calculator designed for instantaneous velocity or a more versatile calculator capable of handling various instantaneous calculations based on the provided input.

## Instantaneous Velocity and Acceleration:

The relationship between instantaneous velocity and acceleration is described by calculus. Instantaneous velocity represents the rate of change of an object’s position with respect to time at a specific instant. Acceleration, on the other hand, represents the rate of change of an object’s velocity with respect to time.

To find the instantaneous velocity from acceleration, you need to integrate the acceleration function with respect to time. Conversely, to find the instantaneous acceleration from velocity, you need to differentiate the velocity function with respect to time.

The specific calculation of instantaneous velocity and acceleration depends on the given information, such as the position function, velocity function, or acceleration function of the object’s motion.

## Conclusion

• It gives the velocity at any point in time during a particle’s motion. Instantaneous velocity is a continuous function of time. The instantaneous velocity at a certain time can be calculated by taking the derivative of the position function, which gives us the functional form of instantaneous velocity $$v(t)$$.
• Instantaneous velocity is a vector that can be negative.
• Instantaneous speed is determined by taking the absolute value of instantaneous velocity, which is always positive.
• The average speed is calculated by dividing the distance traveled by the elapsed time.
• The slope of a position-versus-time graph at a given time indicates the instantaneous velocity.

## FAQs

What are instantaneous velocity and average velocity?

Average velocity is defined as the ratio of total displacement done by the body to the time taken by the body. While instantaneous velocity is defined as the velocity of a body at a specific point of time i.e. displacement of a body at a specific point of time.

Is instantaneous velocity constant?

Instantaneous velocity is the velocity of an object at a given point in time. Objects moving with constant velocity have the same instantaneous velocity, average velocity, and constant velocity.

What is the difference between instantaneous velocity and velocity?

Average velocity is defined as a change in position (or displacement) over the course of a journey, while instantaneous velocity is the velocity of an object at a given moment in time and space as determined by the slope of a tangent line. The terms “speed” and “velocity” are interchangeable in everyday usage.

Why do we use instantaneous velocity?

Instantaneous velocity is the speed of an object at a single instant in time. It would be the same average velocity and instantaneous velocity if the object was moving at a constant speed. In all other cases, they are unlikely to be the same.

How do you find instantaneous velocity in calculus?

Calculating an object’s velocity at any point along its path is possible with calculus. Instantaneous velocity is defined by the equation v = (ds)/(dt); or, in other words, the derivative of the average velocity equation.

How To Find an Instantaneous Velocity Calculator?

To find an instantaneous velocity calculator, you can search online using search engines or visit websites that offer mathematical or physics calculators. Look for calculators that specifically mention instantaneous velocity calculation or velocity at a specific instant in time. There are various online platforms and websites that provide these calculators for free. You can also consider using scientific or graphing calculators that have built-in functions for calculating derivatives or instantaneous velocity.

What Is The Instantaneous Velocity At 7 Seconds?

To determine the instantaneous velocity at 7 seconds, you need to have additional information, such as the object’s position function, velocity function, or acceleration function. Without this information, it is not possible to provide a specific value for the instantaneous velocity at 7 seconds.

If you have the position function of the object, you can differentiate it with respect to time to obtain the velocity function. Then, you can evaluate the velocity function at t = 7 seconds to find the instantaneous velocity at that time. Similarly, if you have the velocity function or acceleration function, you can evaluate them at t = 7 seconds to determine the instantaneous velocity.

How Do I Calculate Instantaneous Velocity?

To calculate the instantaneous velocity, you can follow these steps:

1. Determine the information available: You need to know the position function, velocity function, or acceleration function of the object’s motion. If you have the position function, proceed to step 2. If you have the velocity function or acceleration function, skip to step 4.

2. Differentiate the position function: If you have the position function, take the derivative of the function with respect to time. This will give you the velocity function.

3. Evaluate the velocity function: Once you have the velocity function, plug in the desired time value into the function to find the instantaneous velocity at that time.

4. Use the given information: If you have the velocity function or acceleration function, skip steps 2 and 3 and directly evaluate the function at the desired time value to find the instantaneous velocity.

It’s important to note that the exact method of calculating instantaneous velocity depends on the specific information provided, such as the function or data points given. Calculating instantaneous velocity may involve calculus techniques, such as differentiation, or evaluating functions at specific time values.