Instantaneous Velocity Formula

Last Updated : 04 Feb, 2024

The speed of a moving item at a given point in time while retaining a specific direction is known as instantaneous velocity. With the passage of time, the velocity of an object changes. On the other hand, velocity is defined as the ratio of change in position to change in time when the difference in time becomes zero. Let’s have a look at the idea of instantaneous velocity.

Instantaneous Velocity

The velocity of a moving item at a given point in time is called instantaneous velocity. The rate of change of location for a very short time span, i.e., almost zero, is referred to as instantaneous velocity. The SI unit of m/s is used to measure it. In addition, the magnitude of instantaneous velocity is instantaneous speed. It has the same value as instantaneous velocity but lacks direction. The instantaneous and standard velocity of an item with uniform velocity may be the same.

Instantaneous Velocity Formula

To determine the instantaneous velocity of a particular body at any given time, the Instantaneous Velocity Formula is used. As follows:

Instantaneous Velocity $=\lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$

Where,

Δt = Small time Interval,
x = Displacement,
t = Time.

It’s a quantity that has a vector. The slope of a distance-time graph, or x-t graph, can also be used to determine it.

Derivation of Instantaneous Formula

The velocity between two points in a temporal limit where the time between them eventually becomes 0 is known as instantaneous velocity. The position of x in relation to the function of t is represented by X(t). The average velocity between the two places can be calculated using the following equation:

$V=\frac{x(t_{2})-x(t_{1})}{(t_{2}-t_{1})}$

Let, t₁ = t and t₂= t + Δt

The Δt must be 0 in order to compute instantaneous velocity. Using the limit and the expressions,

Instantaneous velocity $=\lim_{\Delta t\rightarrow 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$

Therefore, Instantaneous Velocity $=\frac{dx(t)}{dt}$

Sample Questions

Question 1: Explain the Concept of the Instantaneous Velocity formula in brief.

Answer:

Instantaneous velocity is defined as the rate at which a position changes over a short time interval. With the exception of having no direction, instantaneous velocity is comparable to instantaneous speed. As a result, instantaneous velocity is defined as the speed of a moving object at a certain point in time.

The speedometer needle, which indicates the car’s speed every hour, varies. Instantaneous velocity refers to this fluctuation, as well as the direction of the car, over a given duration.

Question 2: With a function x = 9t² + t + 7, a given item moves in a straight line for time (t) = 2s. Calculate the Instantaneous velocity in the present moment.

Solution:

Since, Instantaneous Velocity = dx/dt

∴ Instantaneous Velocity = d(9t²+ t + 7)/dt

∴ Instantaneous Velocity = 18t + 1

t = 2s ⇢ (Given)

V(t) = 18t + 1

∴ V(3) = 18(3) + 1

∴ V(3) = 55 m/s

Question 3: When the position of the supplied particle is x(t) = 6t + 0.1t² m at t= 3.8s, calculate the instantaneous velocity.

Solution:

Since, Instantaneous Velocity = dx/dt

∴ Instantaneous Velocity = d(6t + 0.1t²)/dt

∴ Instantaneous Velocity = 0.2t + 6

t = 3.8s, V(t) = 0.2t + 6 = 0.2(3.8) + 6

∴ V(3.8) = 6.76 m/s

Question 4: S(t) = 2t³ + 9t, which travels for 15 seconds before smashing, is the equation of motion for a bullet traveling in a straight path. Calculate the instantaneous velocity during an 8-second timeframe.

Solution:

Since, Instantaneous Velocity = ds/dt

∴ Instantaneous Velocity = d(2t³ + 9t)/dt

∴ Instantaneous Velocity = 6t²+ 9

t = 8s ⇢ (Given)

S(t) = 6t²+ 9

∴ S(8) = 6(8)²+ 9

∴ S(8) = 393 m/s

Question 5: S(t) = 10t² + 7 is an approximate equation of motion for a body moving under gravity. 4 seconds after the release, compute the instantaneous velocity.

Solution:

Since, Instantaneous Velocity = ds/dt = d(10t² + 7)/dt

∴ Instantaneous Velocity = 20t

t = 4s ⇢ (Given)

S(t) = 20t = 20(4)

∴ S(4) = 80 m/s

Question 6: x(t) = 8t + 3t² m calculates an object’s position. calculate the average velocity between 4s and 6s and the instantaneous velocity at t = 2.0s.

Solution:

Since, Instantaneous Velocity = ds/dt = d(8t + 3t²)/dt

∴ Instantaneous Velocity = 8 + 6t

t = 2.0s ⇢ (Given)

V(t) = 8 + 6t = 8 + 6(2.0)

∴ V(2.0) = 20 m/s

We determine the values of x(4s) and x(6s) for average velocity between 4 and 6 s:

∴ X(4) = 8(4) + 3(4)² = 32 + 48 = 80 m

∴ X(6) = 8(6) + 3(6)² = 48 + 108 = 156 m

Final average velocity,

V = 156 – 3.5 × 0 – 4

∴ V = 152 m/s