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

Instantaneous Velocity Calculator

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 \(t_0\).

Instantaneous Velocity Calculator
Instantaneous Velocity 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.

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

Velocity-Time Graph
Velocity-Time Graph

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.

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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.

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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.

Instantaneous Velocity Versus Speed
Instantaneous Velocity Versus Speed

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.

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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.

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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.