Instantaneous Velocity Calculator

Instantaneous Velocity Calculator

This instantaneous velocity calculator is used to calculate the velocity of an object at a particular instance for the given displacement and time. 

What is Instantaneous Velocity?

"A velocity of an object at a given moment in time"

When an object moves with a uniform velocity, then its instantaneous velocity will be equal to the standard velocity. 

How to Calculate Instantaneous Velocity?

It can be calculated using a formula that shows the instantaneous velocity which is the limit of the average velocity when the time interval tends towards zero. 

Instantaneous Velocity Formula:

The instantaneous velocity is defined as the limit of the average velocity as the time interval approaches zero:

$$ v_\text{inst} = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t} $$

Or, using derivatives:

$$ v_\text{inst} = \frac{dx(t)}{dt} $$

Where:

• \(\Delta t\) = infinitesimally small interval of time
• \(x\) = displacement of the object
• \(t\) = time
• \(dx/dt\) = derivative of the position function with respect to time

Example:

Suppose a car is moving along a straight path. Find its instantaneous velocity at t = 3 s.

The car's position function is:

x(t) = 5t² + 2t + 4

Solution:

Step 1: Use the instantaneous velocity formula:

$$ v_\text{inst} = \frac{dx(t)}{dt} $$

Step 2: Take the derivative of the position function:

$$ v_\text{inst} = \frac{d}{dt}(5t^2 + 2t + 4) $$

Step 3: Simplify the derivative:

$$ v_\text{inst} = 10t + 2 $$

Step 4: Substitute t = 3 s:

$$ v_\text{inst} = 10(3) + 2 = 32 \, \text{m/s} $$

Hence, at 3 seconds, the instantaneous velocity of the car is 32 m/s. This means the car's position is changing at a rate of 32 meters every second at that specific moment.

Difference between Instantaneous Speed & Instantaneous Velocity:

➥ Instantaneous Speed:

Instantaneous speed is said to be the magnitude of instantaneous velocity

• Scalar quantity
• Formula of instantaneous speed is: 

Speedint = ds dt

➥ Instantaneous Velocity:

Instantaneous velocity is the change of position which takes place at a small change in time

• Vector quantity
• Formula of instantaneous velocity is: 

Vint = dx dt