Equations of Motion

The equation of motion is a formula that links the net force acting on an object to its acceleration. In high school physics, for an object with a constant mass, we write it as:

$\sum F = ma$

When searching for "equations of motion," the first thing to keep in mind is that $F=ma$ doesn't simply mean "if there is a force, the velocity changes," but rather that "if there is a net force, acceleration is produced."

The direction of acceleration is the same as the direction of the net force. Additionally, the larger the mass $m$, the smaller the acceleration for the same net force. In other words, the equation of motion is the rule that determines "in which direction and by how much the velocity changes."

What does the equation of motion actually represent?

The core of the equation of motion is that force does not determine velocity directly, but rather determines the change in velocity. A force creates acceleration, and if that acceleration persists, the velocity changes.

If the net force is $0$, then:

$\sum F = 0 \quad \Rightarrow \quad a = 0$

In this case, the object will either remain at rest or continue in uniform linear motion. Even if acceleration is $0$, it doesn't necessarily mean that the velocity is $0$.

When can you use $F=ma$?

This form of the $F=ma$ is used in problems where mass can be considered constant. Most high school physics problems involving carts, inclined planes, free fall, and springs can be handled under this condition.

On the other hand, in scenarios where mass changes—such as a rocket consuming fuel—it is safer not to rely solely on $F=ma$. Before applying it, always check: "Can I treat the mass as constant in this problem?"

How to set up the equation of motion

1. Select a single object.
2. Draw a diagram of the forces acting on that object.
3. Define the positive direction.
4. Calculate the net force for each direction.
5. Set up $\sum F = ma$ to find the acceleration.

The key is not to plug numbers into $ma$ right from the start. Organize the forces first, and then set up the equation using the remaining net force. For inclined planes or 2D motion, the basic approach is to break the forces into components and consider the $x$ direction and $y$ direction separately.

Example: Finding acceleration on a horizontal surface with friction

Suppose an object of mass $2\,\mathrm{kg}$ is pulled to the right with a force $10\,\mathrm{N}$. A frictional force $4\,\mathrm{N}$ acts to the left. Assume the mass is constant.

Setting the rightward direction as positive, the net force is:

$\sum F = 10 - 4 = 6\,\mathrm{N}$

Applying the equation of motion here, we get:

$6 = 2a$

Therefore:

$a = 3\,\mathrm{m/s^2}$

The acceleration is to the right because the net force is to the right.

The point of this example is not to use $10\,\mathrm{N}$ as is. You must account for friction and use the resulting net force $6\,\mathrm{N}$. If you get stuck with the equation of motion, remember: "What goes into the equation is the net force, not a single individual force."

Common mistakes with the equation of motion

Using a single force instead of the net force

This is the most common error. What you plug into the equation of motion is the net force, not just the pushing force or gravity alone.

Writing the equation without defining a direction

If you don't decide whether right or left is positive, your signs will be inconsistent. Once you define the direction, you simply enter forces acting in the opposite direction as negative.

Confusing the equation of motion with constant acceleration formulas

Equations like:

$v = v_0 + at,\qquad x = x_0 + v_0 t + \frac{1}{2}at^2$

are formulas for uniformly accelerated motion. These are not the equation of motion itself. First, use the equation of motion to find $a$, and then proceed to velocity or position formulas if necessary.

Thinking of equilibrium and motion as different things

Whether an object is stationary or moving at a constant speed, if the acceleration is $0$, then:

$\sum F = 0$

Equilibrium is simply a specific case within the equation of motion.

When to use the equation of motion

Use the equation of motion whenever you want to determine motion from forces—such as with carts, inclined planes, elevators, falling objects, or circular motion. It is the starting point for any problem where the goal is to "first find the acceleration."

It becomes even more powerful when combined with a Free Body Diagram (FBD). Organizing forces visually before writing the equation significantly reduces sign errors and missing forces.

How to remember: "Net Force $\rightarrow$ Acceleration $\rightarrow$ Change in Velocity"

It is easier to organize your thinking if you view it as: "The net force determines the acceleration, and that acceleration leads to changes in velocity and position." If you try to determine the velocity immediately from the force, you skip the intermediate step of acceleration and are more likely to get confused.

Try this next

Using the same example, try calculating for yourself how the acceleration changes if you change the pulling force to $14\,\mathrm{N}$. By comparing the results after changing just one value, the role of the equation of motion becomes much clearer.