Impulse and Momentum — Definition

Definition

Imagine you're playing billiards. When the cue stick hits the ball, the ball starts moving. How 'much' motion it has depends on its mass and how fast it's going. This 'quantity of motion' is what we call momentum.

Mathematically, momentum (denoted by $p$) is simply the product of an object's mass ($m$) and its velocity ($v$). Since velocity is a vector (it has both magnitude and direction), momentum is also a vector quantity, pointing in the same direction as the velocity.

Its SI unit is kilogram-meter per second (kg\cdot m/s).

Now, think about the impact of the cue stick on the ball. The stick applies a force to the ball, but only for a very short time. The effect of this force over that time is what we call impulse. Impulse (denoted by $I$) is defined as the product of the average force ($F_{avg}$) acting on an object and the time interval ($\Delta t$) over which the force acts.

So, $I = F_{avg} \Delta t$. Like force, impulse is also a vector quantity, and its direction is the same as the direction of the force. Its SI unit is Newton-second (N\cdot s), which you'll find is dimensionally equivalent to kg\cdot m/s.

The most beautiful connection between these two concepts is the Impulse-Momentum Theorem. This theorem states that the impulse applied to an object is equal to the change in its momentum. In simpler terms, if you apply an impulse to an object, you're changing how much 'motion' it has.

So, $I = \Delta p = p_{final} - p_{initial}$. This means if you want to change an object's momentum, you need to apply an impulse. A large force for a short time, or a small force for a long time, can produce the same impulse and thus the same change in momentum.

This theorem is incredibly useful for analyzing collisions, impacts, and other situations where forces are not constant or act for very brief periods, making direct application of Newton's second law ($F=ma$) difficult.