Q.Define the Instantaneous velocity and how is it calculated?
🔒You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Instantaneous Velocity
Instantaneous Velocity: From "How Fast" to "How Fast Right Now"
You already know average velocity. If a car travels 120 km in 2 hours, its average velocity is 60 km/h. That tells you the overall rate, but it hides everything that happened in between — the traffic jams, the sudden bursts of speed, the moments the car was completely stopped.
Now imagine you want to know the car's velocity at exactly 10:15 AM, not averaged over an hour or a minute. That's instantaneous velocity — the velocity at a single instant of time.
The Intuition: Zooming In
Think of a speedometer needle. When you drive, the needle doesn't stay fixed at 60 km/h. It jumps up when you accelerate, drops when you brake. At any given moment, the needle points to a specific number. That number is your instantaneous speed (velocity, if direction matters).
But here's the puzzle: at a single instant, the car hasn't moved any distance. How can you have a speed if Δt=0? You can't divide by zero.
The trick is to shrink the time interval smaller and smaller, and see what the average velocity approaches.
The Precise Definition
Let s(t) be the position of an object at time t. The average velocity over a time interval [t,t+h] is:
vavg=hs(t+h)−s(t)
Now, let h get closer and closer to 0 (but never equal to 0). If the average velocity settles down to a single number as h→0, that number is the instantaneous velocity at time t:
v(t)=limh→0hs(t+h)−s(t)
v(t)=limh→0hs(t+h)−s(t)
This limit is exactly the derivative of position with respect to time. In calculus notation: v(t)=s′(t).
A Concrete Example
Suppose a ball is dropped from rest, and its height (in meters) after t seconds is s(t)=4.9t2 (ignoring air resistance).
Average velocity from t=2 to t=2.1 seconds:
vavg=0.14.9(2.1)2−4.9(2)2=0.14.9(4.41−4)=0.14.9×0.41=20.09 m/s
Average velocity from t=2 to t=2.01:
vavg=0.014.9(2.01)2−4.9(2)2=0.014.9(4.0401−4)=19.649 m/s
Average velocity from t=2 to t=2.001:
vavg=0.0014.9(2.001)2−4.9(2)2=19.6049 m/s
The numbers are converging to 19.6 m/s. That's the instantaneous velocity at t=2 seconds.
Using the derivative: v(t)=9.8t, so v(2)=19.6 m/s. Matches perfectly.
Key Takeaways for Exams
| Concept | Meaning | Formula |
|---|---|---|
| Average velocity | Total displacement ÷ total time | ΔtΔs |
| Instantaneous velocity | Velocity at a single moment | limh→0hs(t+h)−s(t) |
Instantaneous velocity is the velocity of an object at one specific instant of time, defined as the limiting value of average velocity as the time interval considered shrinks to zero; it is calculated by differentiating the object's position with respect to time. …
Instantaneous velocity is the rate of change of position at one particular instant, calculated as the derivative dx/dt.
Definition: The instantaneous velocity of an object at a given instant of time is the limit of its average velocity as the time interval Δt over which the average is taken becomes infinitesimally small (approaches zero).
If x is the position of the object at time t, and it moves to position x + Δx at time t + Δt, the average velocity over this interval is:
v(avg) = Δx/Δt
Instantaneous velocity is obtained by taking the limit Δt → 0:
v = lim(Δt → 0) Δx/Δt = dx/dt
…
- CBSE 2026Set ANNUAL1 markMCQQ.The position of an object moving along x-axis is given by the equation x = 5t - t^2 where x is in m and t in s. The instantaneous speed at t = 2s is :(a) 6 m/s(b) 1 m/s(c) 14 m/s(d) 3 m/s
›Reveal solutionSolution
Differentiating x = 5t - t^2 gives v = 5 - 2t; substituting t = 2 s gives instantaneous speed = 1 m/s.
Given: x = 5t - t^2 (x in m, t in s).
Instantaneous velocity is defined as v = dx/dt.
Differentiating:
v = d/dt (5t - t^2) = 5 - 2t
At t = 2 s:
v = 5 - 2(2) = 5 - 4 = 1 m/s
…
- CBSE 2024Set ANNUAL1 markMCQQ.If the displacement-time graph of a particle is represented by y = mx + c, then the particle is moving with (A) Constant speed (B) Constant velocity (C) Variable velocity (D) Constant momentum
›Reveal solutionSolution
A linear x-t graph y=mx+c means constant velocity.
Velocity is the slope of the displacement-time graph, v=dy/dx (treating x as time here). For y=mx+c, the slope is m everywhere — it never changes with time, so the particle moves with constant velo …
- CBSE 2023Set ANNUAL1 markMCQQ.The motion of a particle is described by the equation x = at + bt^2, where a = 15 cms^-1 and b = 3 cms^-2. Its instantaneous velocity at t = 3 s will be(1) 33 cms^-1(2) 18 cms^-1(3) 16 cms^-1(4) 32 cms^-1
›Reveal solutionSolution
Instantaneous velocity is v = dx/dt; differentiating the given position equation and substituting t = 3 s gives 33 cm/s.
Given: x = at + bt^2, with a = 15 cm s^-1, b = 3 cm s^-2.
Instantaneous velocity: …
- CBSE 2023Set ANNUAL1 markMCQQ.Slope of displacement-time graph in straight line motion represents:(a) time(b) distance(c) velocity(d) acceleration
›Reveal solutionSolution
The slope of an x-t graph is dx/dt, which is exactly how velocity is defined.
For motion in a straight line, displacement x is plotted against time t. Velocity is defined as the rate of change of displacement:
v=dtdx
Graphically, dx/dt at any point of the curve is the slope of the tangent to the x-t graph at that point. For uniform motion the graph is a straight line and its constant slope gives the (constant) velocity; for non-uniform motion the slope changes from point to point and give …
- CBSE 2021Set sz1 markQ.Define instantaneous velocity.
›Reveal solutionSolution
Instantaneous velocity is the limiting value of average velocity as the time interval considered tends to zero; it equals dx/dt.
Average velocity over a time interval delta-t is defined as delta-x / delta-t, the displacement divided by the time taken. As delta-t is made smaller and smaller (delta-t -> 0), the average velocity approaches a fixed limiting value, which is called the instantaneous velocity at that instant:
v = lim(delta-t -> 0) (delta-x/delta-t) = dx/dt
…
- CBSE 2020Set ANNUAL1 markMCQQ.The position of an object moving along x-axis is given by, x = a + bt^2 where a = 2m and b = 3 m/s^2 and t is measured in seconds. The velocity at t = 2s is(a) 2 m/s(b) 6 m/s(c) 12 m/s(d) 14 m/s
›Reveal solutionSolution
Differentiating x = a + bt^2 gives v = 2bt = 6t (since b = 3), which evaluates to 12 m/s at t = 2 s.
Given: x = a + b t^2, with a = 2 m, b = 3 m/s^2
…
- CBSE 2017Set ANNUAL1 markQ.What does the slope of displacement-time graph represent?
›Reveal solutionSolution
The slope of an x-t graph at any instant equals the particle's instantaneous velocity at that instant.
For a particle moving along a straight line, if x is its position at time t, the average velocity between two close instants t and t+Δt is v_avg = Δx/Δt, which is exactly the slope of the chord joining the two corresponding points on the x-t graph. As Δt is made smaller and smaller (Δt → 0), the chord becomes the tangent to the curve at that point, and the average velocity becomes the instantaneous velocity: v = dx/dt. So at every point on an x-t graph, the slope of the tangent drawn at that point gives the instantaneous velocity of the particle at that instant …
🎓Unlock everything free for 14 days
- ✓Full step-by-step solutions
- ✓Concept-first explanations
- ✓Methods, shortcuts & mistakes
- ✓PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.