KAHIBARO
Table of Contents
Force as the Rate of Change of Momentum
Momentum and force are closely connected. Momentum tells us how much motion an object has, taking into account both its mass and its velocity. Force describes how that motion changes. The key idea of this chapter is that force is not only related to acceleration, but more generally to the change of momentum.
For a single object, linear momentum is defined by
$$
\vec{p} = m\vec{v}
$$
where $\vec{p}$ is momentum, $m$ is mass, and $\vec{v}$ is velocity. Since velocity is a vector, momentum is also a vector. A change in speed, a change in direction, or both, will change momentum.
The relation between force and momentum is
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}
$$
This means that the net force on an object equals the rate at which its momentum changes with time.
Important general law:
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}
$$
A net force does not simply mean motion. It means a change in momentum.
Connection to Newton's Second Law
For many beginner problems, the mass of an object stays constant. In that case,
$$
\vec{p} = m\vec{v}
$$
so
$$
\frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt} = m\frac{d\vec{v}}{dt} = m\vec{a}
$$
Therefore, when mass is constant,
$$
\vec{F}_{\text{net}} = m\vec{a}
$$
This is the familiar form of Newton's second law. So $ \vec{F} = m\vec{a} $ is actually a special case of the more general momentum form.
If mass is constant:
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} = m\vec{a}
$$
If mass is not constant, use the momentum form directly.
What Force Really Changes
Because momentum depends on velocity, force can affect motion in different ways. A force can increase momentum, decrease momentum, or change its direction.
If a force acts in the same direction as the motion, the magnitude of momentum increases. If it acts opposite to the motion, the magnitude of momentum decreases. If it acts sideways, the direction of momentum changes.
This is why an object moving in a circle can have a changing momentum even if its speed stays constant. The direction of the velocity changes, so the momentum changes too.
Units of Force and Momentum
Momentum has SI units
$$
\mathrm{kg \cdot m/s}
$$
Force has SI units
$$
\mathrm{N} = \mathrm{kg \cdot m/s^2}
$$
The momentum form of force is consistent with these units, because dividing momentum by time gives
$$
\frac{\mathrm{kg \cdot m/s}}{\mathrm{s}} = \mathrm{kg \cdot m/s^2} = \mathrm{N}
$$
The following table compares the two quantities.
| Quantity | Symbol | Definition | SI unit |
|---|---|---|---|
| Momentum | $\vec{p}$ | $m\vec{v}$ | $\mathrm{kg \cdot m/s}$ |
| Net force | $\vec{F}_{\text{net}}$ | $\dfrac{d\vec{p}}{dt}$ | $\mathrm{N}$ |
Constant Force and Momentum Change
If the net force is constant and acts over a time interval $\Delta t$, then the change in momentum is
$$
\Delta \vec{p} = \vec{F}_{\text{net}} \Delta t
$$
This equation is very useful because it directly links force and momentum change.
If the force acts longer, the momentum changes more. If the force is larger, the momentum also changes more.
This idea leads naturally to impulse, which is treated in its own chapter. Here, the important point is simply that force applied over time changes momentum.
For constant net force:
$$
\Delta \vec{p} = \vec{F}_{\text{net}} \Delta t
$$
A larger force or a longer time gives a larger change in momentum.
Example with Constant Mass
Suppose a cart of mass $2\,\mathrm{kg}$ moves at $3\,\mathrm{m/s}$ to the right. Its initial momentum is
$$
p_i = mv = (2)(3) = 6\,\mathrm{kg \cdot m/s}
$$
If a net force of $4\,\mathrm{N}$ acts to the right for $2\,\mathrm{s}$, then the change in momentum is
$$
\Delta p = F\Delta t = (4)(2) = 8\,\mathrm{kg \cdot m/s}
$$
So the final momentum is
$$
p_f = p_i + \Delta p = 6 + 8 = 14\,\mathrm{kg \cdot m/s}
$$
The final velocity is
$$
v_f = \frac{p_f}{m} = \frac{14}{2} = 7\,\mathrm{m/s}
$$
This example shows how force changes momentum directly.
Direction Matters
Since momentum and force are vectors, signs or directions must be handled carefully. In one dimension, we often choose one direction as positive.
If right is positive, then a force to the left is negative. A momentum to the left is also negative. This makes the equations easier to use.
For example, if an object has momentum $+10\,\mathrm{kg \cdot m/s}$ and a force produces a momentum change of $-15\,\mathrm{kg \cdot m/s}$, then the final momentum is
$$
p_f = 10 - 15 = -5\,\mathrm{kg \cdot m/s}
$$
The negative sign means the object is now moving to the left.
A Graphical View
A graph of momentum versus time helps show the meaning of force. Since
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}
$$
the slope of a momentum time graph is the net force.
A steep slope means a large force. A zero slope means zero net force, so momentum stays constant.
If the graph is a straight line, the force is constant. If the slope changes, the force changes.
Variable Mass Situations
In some situations, mass changes with time. A rocket is a common example, because it burns fuel as it moves. In such cases, the simple form $F = ma$ is not enough by itself. The general momentum law must be used:
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}
$$
This chapter does not go further into variable mass systems, but it is important to know that the momentum form is more fundamental.
Physical Meaning
The momentum form of force gives a deep view of motion. It says that force is not directly tied to velocity, but to change in momentum. An object can have large momentum and still feel no net force, as long as that momentum remains constant. On the other hand, even if an object's speed stays the same, a force may still act if the direction of momentum changes.
This helps explain many situations in mechanics, especially collisions and interactions over short times.
Summary Relations
The central equations of this chapter are:
$$
\vec{p} = m\vec{v}
$$
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}
$$
and for constant mass,
$$
\vec{F}_{\text{net}} = m\vec{a}
$$
and for constant force over a time interval,
$$
\Delta \vec{p} = \vec{F}_{\text{net}} \Delta t
$$
Core ideas to remember:
$$
\vec{p} = m\vec{v}
$$
$$
\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}
$$
For constant mass:
$$
\vec{F}_{\text{net}} = m\vec{a}
$$
Force changes momentum, not simply motion.