KAHIBARO
Table of Contents
Meaning of Momentum
Momentum is a quantity that tells us how much motion an object has. In classical mechanics, the linear momentum of an object depends on two things, its mass and its velocity. A heavy object moving slowly can have a lot of momentum, and a light object moving quickly can also have a lot of momentum.
The symbol for linear momentum is usually $\vec{p}$. It is defined by
$$
\vec{p} = m\vec{v}
$$
where $m$ is the mass of the object and $\vec{v}$ is its velocity.
Because velocity is a vector, momentum is also a vector. This means momentum has both magnitude and direction. The direction of the momentum is the same as the direction of the velocity.
Important definition:
$$
\vec{p} = m\vec{v}
$$
Momentum is a vector quantity.
Its direction is the same as the direction of velocity.
Why Momentum Matters
Momentum is useful because it describes motion in a way that becomes especially powerful when studying interactions between objects, such as pushes, impacts, and collisions. Position and velocity describe how one object moves. Momentum helps describe what happens when objects affect each other.
For example, a rolling truck is harder to stop than a rolling bicycle moving at the same speed. The truck has a larger mass, so it has more momentum. A fast tennis ball can also be hard to catch because its velocity is large, even though its mass is small.
In everyday language, people often say that something has "a lot of momentum" when it is hard to stop. Physics gives this idea a precise mathematical meaning.
Scalar Form in One Dimension
In one dimensional motion, momentum can be written without arrows if we keep track of sign. Then we write
$$
p = mv
$$
If the object moves in the positive direction, $p$ is positive. If it moves in the negative direction, $p$ is negative.
This sign is important. Two objects can have the same speed but opposite momenta if they move in opposite directions.
Consider an object of mass $2\,\text{kg}$.
If it moves at $3\,\text{m/s}$ to the right,
$$
p = (2)(3) = 6\,\text{kg m/s}
$$
If it moves at $3\,\text{m/s}$ to the left, taking left as negative,
$$
p = (2)(-3) = -6\,\text{kg m/s}
$$
The magnitudes are the same, but the directions are opposite.
Units of Momentum
Since momentum is mass times velocity, its SI unit is
$$
\text{kg} \cdot \text{m/s}
$$
There is no special common name for this unit in introductory mechanics, so we usually leave it as kilogram meter per second.
| Quantity | Symbol | SI unit |
|---|---|---|
| Mass | $m$ | $\text{kg}$ |
| Velocity | $\vec{v}$ | $\text{m/s}$ |
| Momentum | $\vec{p}$ | $\text{kg}\cdot\text{m/s}$ |
Momentum and Direction
Because momentum is a vector, changing direction changes momentum, even if the speed stays the same. This is an important idea.
Imagine a car moving east. Its momentum points east. If the car turns and moves north at the same speed, the magnitude of its momentum may stay the same, but the momentum itself has changed because its direction has changed.
In two or three dimensions, momentum can be written in components:
$$
\vec{p} = m\vec{v}
$$
If
$$
\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}
$$
then
$$
\vec{p} = mv_x \hat{i} + mv_y \hat{j} + mv_z \hat{k}
$$
So the momentum components are
$$
p_x = mv_x,\qquad p_y = mv_y,\qquad p_z = mv_z
$$
Examples
A $5\,\text{kg}$ cart moving at $4\,\text{m/s}$ has momentum
$$
p = mv = (5)(4) = 20\,\text{kg}\cdot\text{m/s}
$$
If a $0.20\,\text{kg}$ ball moves at $15\,\text{m/s}$, then
$$
p = (0.20)(15) = 3.0\,\text{kg}\cdot\text{m/s}
$$
Even though the ball is fast, its momentum is less than that of the cart because its mass is much smaller.
Now consider a $1000\,\text{kg}$ car moving at $20\,\text{m/s}$:
$$
p = (1000)(20) = 2.0 \times 10^4\,\text{kg}\cdot\text{m/s}
$$
This much larger momentum helps explain why moving cars are difficult to stop quickly.
Rest and Zero Momentum
If an object is at rest relative to the chosen reference frame, then its velocity is zero. Therefore its momentum is also zero:
$$
\vec{v} = 0 \quad \Rightarrow \quad \vec{p} = 0
$$
This shows that momentum depends on the reference frame. An object may be at rest in one frame and moving in another, so its momentum can be different for different observers.
Comparing Momentum of Different Objects
Momentum depends on both mass and velocity, so it is not enough to know only one of them. The table below shows this clearly.
| Object | Mass | Velocity | Momentum |
|---|---|---|---|
| Toy car | $0.50\,\text{kg}$ | $2.0\,\text{m/s}$ | $1.0\,\text{kg}\cdot\text{m/s}$ |
| Soccer ball | $0.40\,\text{kg}$ | $10\,\text{m/s}$ | $4.0\,\text{kg}\cdot\text{m/s}$ |
| Bicycle | $15\,\text{kg}$ | $5.0\,\text{m/s}$ | $75\,\text{kg}\cdot\text{m/s}$ |
| Car | $1200\,\text{kg}$ | $25\,\text{m/s}$ | $3.0\times10^4\,\text{kg}\cdot\text{m/s}$ |
A larger mass or a larger speed leads to a larger momentum magnitude. If both increase, the momentum increases even more.
Visualizing Momentum
A momentum vector is drawn as an arrow in the direction of motion. A longer arrow represents a larger magnitude of momentum.
If the object reverses direction, the momentum arrow reverses too.
Momentum as a Property of Motion
Momentum is not a force, and it is not energy. It is a separate physical quantity. Two objects can have the same momentum but different masses and speeds. For example:
$$
(2\,\text{kg})(3\,\text{m/s}) = 6\,\text{kg}\cdot\text{m/s}
$$
and
$$
(1\,\text{kg})(6\,\text{m/s}) = 6\,\text{kg}\cdot\text{m/s}
$$
These objects have equal momentum magnitude, but they are not moving in the same way.
Key Idea to Remember
Momentum combines mass and velocity into one vector quantity that describes motion.
Key rule:
$$
\vec{p} = m\vec{v}
$$
For constant mass, momentum increases when mass increases, when speed increases, or both.
In one dimension, the sign of momentum shows direction.
Momentum will become especially important when we study how forces change motion, and when we study collisions between interacting objects.