Balanced in One Direction, Unbalanced in Another (2.4.4) | AP Physics 1: Algebra Notes

AP Syllabus focus: ‘Forces can be balanced in one direction and unbalanced in another, changing velocity only where force is unbalanced.’

Motion in two dimensions is analysed by separating forces into perpendicular directions. A system can have zero net force along one axis (no acceleration there) while simultaneously accelerating along another axis.

Core Principle: Balance Can Be Direction-Specific

Forces add as vectors, so net force can be zero in one direction even if it is nonzero in another. The key payoff is that acceleration components depend only on the corresponding net force components.

Component Independence (What You’re Allowed to Conclude)

If an object’s forces are balanced in the $x$ -direction, then its $x$ -velocity does not change, even if there is a nonzero net force in the $y$ -direction. The object may still speed up, slow down, or change direction overall—just not due to acceleration in the balanced direction.

Direction-by-Direction “Balance” Language

Balanced in a direction — the net force component along that axis is zero, so the acceleration component along that axis is zero.

A common confusion is thinking “balanced” means “not moving.” Instead, balanced in one direction means constant velocity in that direction (which could be zero or nonzero).

Newton’s Second Law in Perpendicular Directions

When you split forces into components, you write a separate Newton’s second law equation for each axis.

Free-body diagram on an incline with the weight $W$ resolved into components parallel and perpendicular to the surface. The diagram supports the idea that forces can cancel in one direction (often the perpendicular direction, giving $a_\perp=0$ ) while remaining unbalanced in another (parallel direction, giving nonzero acceleration). Source

This is the algebraic basis for “velocity changes only where force is unbalanced.”

$\sum F_x = m a_x$

$\sum F_x$ = net external force component in the $x$ -direction (N)

$m$ = mass of the system (kg)

$a_x$ = acceleration component in the $x$ -direction (m/s $^2$ )

$\sum F_y = m a_y$

$\sum F_y$ = net external force component in the $y$ -direction (N)

$a_y$ = acceleration component in the $y$ -direction (m/s $^2$ )

These equations are independent: setting $\sum F_x=0$ forces $a_x=0$ , but it does not force $a_y=0$ .

Projectile motion decomposed into independent horizontal and vertical motions: the horizontal component has $a_x=0$ so $v_x$ remains constant, while the vertical component has nonzero acceleration so $v_y$ changes. Recombining the components gives a changing velocity vector along a curved trajectory, even though one component can remain constant. Source

What Changes (and What Doesn’t) When One Direction Is Unbalanced

Balanced direction: velocity component is constant

If $\sum F_x = 0$ , then $a_x = 0$
Therefore $v_x$ is constant (could be $0$ or nonzero)
Position in $x$ still changes if $v_x \neq 0$ , but it changes at a constant rate

Unbalanced direction: velocity component changes

If $\sum F_y \neq 0$ , then $a_y \neq 0$
Therefore $v_y$ changes over time
This can change the object’s overall speed and the direction of its velocity vector

Combined effect: path can curve even if one component is constant

An object can move with constant $v_x$ while its $v_y$ changes; the resulting trajectory is not required to be a straight line. The “only where force is unbalanced” statement is specifically about changes in velocity components, not about whether the object’s position changes.

Recognising Typical Force Patterns (Without Overgeneralising)

Many setups naturally produce balance in one axis:

Vertical balance: weight downward and a support force upward (normal force or tension) cancel, so $\sum F_y=0$ while horizontal forces may be unbalanced.
Horizontal balance: leftward and rightward forces cancel, so $\sum F_x=0$ while vertical forces may be unbalanced.

Be careful: a force “existing” in a direction does not imply imbalance. You must compare the sum of all components along that axis.

How to Apply This on Free-Body Diagrams and Equations

Workflow for “balanced in one direction” problems

Choose coordinate axes (often horizontal/vertical, unless given a tilted axis).
Draw a free-body diagram and label all external forces.
Resolve forces into $x$ and $y$ components if needed.
Add components to get $\sum F_x$ and $\sum F_y$ .
Use $\sum F_x = m a_x$ and $\sum F_y = m a_y$ separately.
Interpret:
- If $\sum F_{\text{axis}}=0$ then $a_{\text{axis}}=0$ and that velocity component is constant.
- If $\sum F_{\text{axis}}\neq 0$ then the velocity component changes in that direction.

Common reasoning checkpoints

“Constant speed” is not guaranteed by balance in one axis; only the component is fixed.
If one axis is balanced, you can often solve the other axis without needing details from the balanced one (except for shared quantities like mass).

FAQ

Speed depends on the magnitude of the full velocity vector. If $v_x$ is constant but $v_y$ increases in magnitude, then $| \vec v |$ increases even though one component is unchanged.

Only if the only vertical forces are $N$ upward and $mg$ downward and the vertical acceleration is zero. Extra vertical forces (e.g., an angled pull) change the balance condition.

Yes. $\sum F_x=0$ implies $a_x=0$, not $v_x=0$. The object can have a nonzero constant $v_x$.

Curved motion occurs whenever acceleration is not parallel to velocity. If acceleration acts only in one axis while velocity has components in both axes, the velocity direction changes, producing a curved path.

Common issues include:

using the wrong angle for components
forgetting signs (positive/negative directions)
mixing axes (writing $y$-components in the $x$-equation)
assuming a force must equal another just because they “look opposite” on the diagram

Practice Questions

Q1 (1–3 marks) A puck slides on frictionless ice. The net force in the $x$ -direction is $0\ \text{N}$ , and the net force in the $y$ -direction is $6\ \text{N}$ upward. The mass is $2\ \text{kg}$ . State which velocity component(s) change, and find $a_y$ .

States $v_x$ does not change because $\sum F_x=0$ so $a_x=0$ (1)
States $v_y$ changes because $\sum F_y\neq 0$ (1)
Calculates $a_y=\sum F_y/m = 6/2 = 3\ \text{m s}^{-2}$ upward (1)

Q2 (4–6 marks) A cart moves on a level track. Vertically, the normal force $N$ and weight $mg$ act. Horizontally, a constant pulling force $F$ acts to the right and a smaller constant resistive force $f$ acts to the left. The cart’s initial velocity is purely to the right. Explain, using Newton’s second law components, which acceleration components are zero/nonzero, and describe how the cart’s velocity changes over time.

Identifies vertical forces and states $\sum F_y = N - mg$ (1)
Concludes vertical forces balance ( $N=mg$ ) so $\sum F_y=0$ and $a_y=0$ (1)
States therefore $v_y$ remains constant (and is zero if initially zero) (1)
Writes horizontal net force $\sum F_x = F - f$ (1)
Concludes $a_x=(F-f)/m$ is nonzero to the right (1)
States $v_x$ increases over time (speed increases; direction stays right) because $a_x$ is to the right (1)

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