Instantaneous Velocity

Average velocity tells us how fast the object is moving in a given interval of time, but it does not tell us how fast that thing is moving in a particular instant of time. Instantaneous velocity addresses these issues by defining the rate of motion at a particular instant.

Instantaneous velocity gives the velocity of the object at a particular instant of time during a given interval. The SI unit of it Instantaneous velocity is also m/s . In addition, the magnitude of instantaneous velocity is instantaneous speed. It has the same value as instantaneous velocity but lacks direction.

• For an object moving with a constant velocity, its instantaneous velocity and average velocity are always equal.
• The slope of the tangent at any point on the distance-time graph (x-t graph) gives us the instantaneous velocity.

Instantaneous Velocity Formula

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

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

Where,

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

Solved Examples:

Question 1: Explain the concept of the instantaneous velocity formula in brief.

Solution: 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) = 3s. Calculate the instantaneous velocity in the present moment.

Solution: Instantaneous Velocity = \frac{dx}{dt}

Instantaneous Velocity = \frac{d(9t^2 + t + 7)}{dt}

∴ Instantaneous Velocity = 18t + 1

t = 3s ⇢ (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 = 3.8 s, calculate the instantaneous velocity.

Solution: Instantaneous Velocity= \frac{dx}{dt}

Instantaneous Velocity = \frac{d(6t + 0.1t^2)}{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: Instantaneous Velocity = \frac{dx}{dt}

Instantaneous Velocity = \frac{d(2t^3 + 9t)}{dt}

∴ Instantaneous Velocity = \frac{d(6t^2 + 9)}{dt}

t = 8s ⇢ (Given)

S(t) = 6t² + 9

∴ S(8) = 6(8)² + 9

∴ S(8) = 393 m/s

Question 5: 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: Instantaneous Velocity = \frac{dx}{dt}

Instantaneous Velocity = \frac{d(8t + 3t^2)}{dt}

∴ Instantaneous Velocity = \frac{d(8 + 6t)}{dt}

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

Unsolved Questions

Question 1. The displacement of an object is given by x(t) = t^3 - 6t^2 + 9t
Find the time at which the instantaneous velocity becomes zero.

Question 2. A particle moves along the x-axis such that its position is given by

x(t) = t^3 - 6t^2 + 9t

(a) Find the instantaneous velocity as a function of time.
(b) Determine the time interval during which the particle moves in the negative direction.
(c) Find the total distance travelled between t = 0 and t = 4 sec.

Question 3. Assertion (A): Instantaneous velocity can be zero even when acceleration is non-zero.
Reason (R): Acceleration depends on the rate of change of velocity, not on the value of velocity.

(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true