Q.The position–time graph of a body of mass $2\ \text{kg}$ is a straight line rising from the origin: its position $x$ increases uniformly from $0$ at $t=0$ to $3\ \text{m}$ at $t=4\ \text{s}$, and thereafter $x$ stays constant at $3\ \text{m}$ for $t>4\ \text{s}$. What is the impulse on the body at $t=0\ \text{s}$ and at $t=4\ \text{s}$?

The Intuition: Why Do We Need a "New" Idea?

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.

The Precise Statement

The Impulse-Momentum Theorem says:

The impulse delivered to an object equals the change in its momentum.

In symbols:

Where:

• $\vec{J}$ is the impulse (a vector)
• $\Delta \vec{p}$ is the change in momentum (also a vector)

And since impulse is force times time:

Breaking It Down Piece by Piece

Momentum ($\vec{p}$) is mass times velocity: $\vec{p} = m\vec{v}$. 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 ($\vec{J}$) is the product of the average force and the time interval over which it acts: $\vec{J} = \vec{F}_{\text{avg}} \, \Delta t$.

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.

Why This Matters: Real-World Examples

Catching a ball (soft vs. hard hands)

• Hard hands: $\Delta t$ is small → $\vec{F}_{\text{avg}}$ is large (it hurts)
• Soft hands: $\Delta t$ is large → $\vec{F}_{\text{avg}}$ is small (it's comfortable)
• In both cases, $\Delta \vec{p}$ 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:

Multiply both sides by $dt$:

Integrate over the time interval:

The left side is the impulse (the area under the force-time graph). The right side is the change in momentum. …