Q.The position–time graph of a body of mass 2kg is a straight line rising from the origin: its position x increases uniformly from 0 at t=0 to 3m at t=4s, and thereafter x stays constant at 3m for t>4s. What is the impulse on the body at t=0s and at t=4s?
Imagine you're catching a cricket ball. If you let your hands stay rigid, the ball stings and might bounce off. But if you give with the ball — pulling your hands back as you catch — the catch feels soft and the ball stops gently.
Same ball, same speed, same change in momentum. But the force you feel is completely different. Why?
The answer is time. When you pull your hands back, you increase the time over which the ball slows down. A longer time means a smaller force — even though the total "oomph" needed to stop the ball is the same. That "oomph" is called impulse.
Note
Impulse is not a mysterious new quantity. It's just force multiplied by the time it acts. If you push gently for a long time, or push hard for a short time, you can produce the same effect.
The Precise Statement
The Impulse-Momentum Theorem says:
The impulse delivered to an object equals the change in its momentum.
In symbols:
J=Δp
Where:
J is the impulse (a vector)
Δp is the change in momentum (also a vector)
And since impulse is force times time:
FavgΔt=mvf−mvi
J=FavgΔt=Δp
Breaking It Down Piece by Piece
Momentum (p) is mass times velocity: p=mv. It's a measure of how hard it is to stop a moving object. A truck moving slowly has large momentum; a bullet moving fast has large momentum too.
Impulse (J) is the product of the average force and the time interval over which it acts: J=FavgΔt.
The theorem connects them: the net impulse changes the momentum. If you apply a net force to an object for some time, its momentum changes by exactly that amount.
Watch out
A common mistake is to think impulse is just force. It's force × time. A huge force acting for a tiny time (like a bat hitting a ball) can produce the same impulse as a tiny force acting for a long time (like a gentle push).
Why This Matters: Real-World Examples
Catching a ball (soft vs. hard hands)
Hard hands: Δt is small → Favg is large (it hurts)
Soft hands: Δt is large → Favg is small (it's comfortable)
In both cases, Δp is the same (ball goes from moving to stopped)
Airbags in cars
Without airbag: your head hits the dashboard in ~0.01 s → huge force
With airbag: your head decelerates over ~0.1 s → force is 10 times smaller
Same change in momentum, but the airbag extends the time
A cricket bat hitting a ball
The bat is in contact with the ball for a few milliseconds
The force during that contact is enormous (hundreds of Newtons)
The impulse changes the ball's momentum from one direction to another
The Mathematical Derivation (Short)
Start from Newton's second law:
Fnet=ma=mdtdv
Multiply both sides by dt:
Fnetdt=mdv
Integrate over the time interval:
∫titfFnetdt=m∫vivfdv=mvf−mvi
The left side is the impulse (the area under the force-time graph). The right side is the change in momentum. …
The uniform slope tells us the body travels at a steady 0.75m s−1 from t=0 to t=4s, then rests. Impulse equals the sudden change in momentum, so it is +1.5kg m s−1 at start and −1.5kg m s−1 when it stops.
Concept: impulse = change in momentum
By the impulse–momentum theorem, J=Δp=mΔv. So we need the velocity just before and just after each instant.
Reading the velocity
For 0<t<4s the graph is a straight line, so the velocity is constant and equals the slope:
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2024Set ANNUAL1 markMCQ
Q.Which of the following have same dimensions ?
(a) Specific heat and latent heat
(b) Momentum and impulse
(c) Moment of momentum and moment of inertia
(d) Tension and surface tension
›Reveal solutionSolution
Impulse J = change in momentum Δp = FΔt, so by definition impulse and momentum are dimensionally identical — option (b).
Let's check each option:
Specific heat [L^2 T^-2 K^-1] vs latent heat [L^2 T^-2] — these differ by a factor of K^-1 (temperature), so NOT the same dimensions.
Momentum p = mv has dimensions [M][LT^-1] = [M L T^-1]. Impulse J is defined as the change in momentum caused by a force acting over a time interval: J = FΔt = Δp. Since impulse is literally equal to a change in momentum, it must have exactly the same dimensions as momentum: [M L T^-1]. This pair is dimensionally identical.
…
Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2024Set ANNUAL1 markMCQ
Q.The sudden force acting on an object for a short interval of time is given by -
(a) change in linear momentum
(b) change in mass
(c) change in force
(d) change in acceleration
›Reveal solutionSolution
By the impulse–momentum theorem, a force acting for a short time interval produces an impulse J = FΔt, which equals exactly the change in the object's linear momentum — option (a).
When a large force acts on a body for a very short interval of time (e.g., a bat striking a ball, a hammer striking a nail), it is called an impulsive force. Even though such a force may be difficult to measure directly (since it may vary rapidly during the brief contact), its overall EFFECT is captured by a quantity called impulse, J:
J = F_avg × Δt
By Newton's second law, F = dp/dt, so integrating over the short time interval Δt:
Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2023Set ANNUAL1 mark
Q.What is the impulsive force?
›Reveal solutionSolution
An impulsive force is a large force acting for a very short time, whose effect is measured by the impulse (force × time) it delivers, equal to the change in momentum it produces.
By Newton's second law, force is the rate of change of momentum: F=dtdp. If a very large force F acts for a very small time interval Δt, the product FΔt — called the impulse — can still be finite and measurable, even though F is very large and Δt is very small. Such forces, which act only for a very brief duration but produce a finite change in momentum, are called impulsive forces.