AP Physics 1: Kinematics

Kinematics is the study of motion without worrying about what causes it. In AP Physics 1, it is the gateway to nearly everything that follows in mechanics: forces, energy, momentum, and rotation all rely on the same habits of mind you build here. If you can interpret motion graphs, use kinematic equations appropriately, and break motion into components, you have the tools to analyze real motion in one and two dimensions with confidence.

The core quantities: position, displacement, velocity, and acceleration

Position and displacement

Position tells you where an object is relative to a chosen origin. In one dimension, position is often written as $x$ (meters). Displacement is the change in position: $\Delta x=x_f-x_i$.

Displacement is a vector quantity, meaning it has direction. That is why walking 3 m east and then 3 m west gives a displacement of 0 m, even though you traveled 6 m total.

Velocity

Velocity measures how quickly position changes and in what direction. Average velocity is $v_{avg}=\frac{\Delta x}{\Delta t}$.

Instantaneous velocity is the velocity at a particular moment. Conceptually, it is the slope of the position-time graph at that instant.

A common AP Physics 1 trap is confusing velocity with speed. Speed is the magnitude of velocity and never negative. Velocity can be negative, indicating motion in the negative direction of your coordinate system.

Acceleration

Acceleration measures how quickly velocity changes. Average acceleration is $a_{avg}=\frac{\Delta v}{\Delta t}$.

Acceleration can be positive or negative, and the sign depends on your axis choice. Negative acceleration does not automatically mean “slowing down.” If an object is moving in the negative direction and speeding up, its acceleration is negative too. Slowing down happens when velocity and acceleration have opposite signs.

Reading motion graphs like a physicist

Graphs are not decoration in AP Physics 1. They are data and a model at the same time.

Position-time graphs ($x$ vs. $t$)

On an $x$ vs. $t$ graph:

• The slope represents velocity.
• A steeper slope means greater speed.
• A flat line means zero velocity (object at rest).
• Curvature indicates changing velocity (acceleration).

If the graph is concave up, velocity is increasing; if it is concave down, velocity is decreasing. Students often mistakenly read the height of the graph as “speed.” The vertical value is position, not velocity.

Velocity-time graphs ($v$ vs. $t$)

On a $v$ vs. $t$ graph:

• The slope represents acceleration.
• The area under the curve represents displacement.

That second point is crucial. If velocity is positive, area is positive displacement; if velocity is negative, area counts as negative displacement. On an exam, you may be asked for distance traveled, which is not the same as displacement. Distance is total area counting everything as positive, often requiring you to split the graph where it crosses $v=0$.

Acceleration-time graphs ($a$ vs. $t$)

On an $a$ vs. $t$ graph:

• The area under the curve represents change in velocity, $\Delta v$.

A constant nonzero acceleration produces a velocity-time graph that is a straight line, and a position-time graph that is a curve (specifically quadratic in time).

Kinematic equations (constant acceleration only)

AP Physics 1 emphasizes when you are allowed to use the “big four” kinematic equations: only when acceleration is constant. Many real motions approximate constant acceleration well, such as free fall near Earth’s surface (ignoring air resistance) or a car speeding up steadily.

Common constant-acceleration relations include:

• $v=v_0+at$
• $\Delta x=v_{0}t+\frac{1}{2}a{t}^2$
• $v^2=v_0^2+2a\Delta x$
• $\Delta x=\frac{(v+v_0)}{2}t$

A practical approach is to list what you know and what you need, then choose the equation that connects them. Avoid hunting for the “right” equation by memory alone. Also keep sign conventions consistent: if up is positive, then gravitational acceleration is $a_y=-g$ where $g\approx 9.8{\text{m/s}}^2$.

Motion in two dimensions: vectors and components

Two-dimensional kinematics is not new physics, it is one-dimensional kinematics applied separately along perpendicular axes. The reason it works is that horizontal and vertical motions are independent when the only acceleration is vertical (as in projectile motion without air resistance).

Vector components

A vector like velocity can be resolved into $x$ and $y$ components:

• $v_x=v\cos \theta$
• $v_y=v\sin \theta$

Similarly, displacement splits into $\Delta x$ and $\Delta y$. In 2D problems, you almost always solve for $x$ and $y$ separately and then combine results if needed (for example, to find a final speed).

Independence of perpendicular motion

If acceleration is zero in the $x$ direction, then $v_x$ is constant. Meanwhile, if acceleration is constant in the $y$ direction (like $-g$), then $y$ motion follows the same constant-acceleration equations you used in one dimension.

This is why, in ideal projectile motion:

• $a_x=0$
• $a_y=-g$

Projectile motion: the flagship 2D application

Projectile motion is the motion of an object launched into the air and moving under gravity alone. The defining feature is that the object keeps its horizontal velocity while its vertical velocity changes linearly due to gravity.

Typical analysis workflow

1. Choose axes (often $+x$ horizontal, $+y$ upward).
2. Break initial velocity into components $v_{0x}$ and $v_{0y}$.
3. Use $x$-motion with $a_x=0$:

• $\Delta x=v_{0x}t$

1. Use $y$-motion with $a_y=-g$:

• $\Delta y=v_{0y}t-\frac{1}{2}g{t}^2$
• $v_y=v_{0y}-gt$

Time is the bridge between the two directions. You often solve for time using the vertical equation, then use that time in the horizontal equation to find range.

Key facts that show up on exams

• At the top of the trajectory, $v_y=0$, but acceleration is still $-g$.
• If a projectile lands at the same height it was launched (symmetric flight), time up equals time down, and the speed at landing equals the launch speed (direction differs).
• Horizontal velocity does not change (again, only in the ideal model with negligible air resistance).

Common pitfalls

• Plugging $g$ in with the wrong sign. Decide your positive direction first.
• Assuming $v=0$ at the peak. Only the vertical component is zero.
• Using one-dimensional equations with mixed components (for example, using $\Delta x$ in an equation meant for vertical motion).

Problem-solving habits that pay off

Use units as a reality check

If your displacement is in meters and time in seconds, then velocity should be in m/s, and acceleration in m/s². Unit mismatches are early warnings of setup errors.

Sketch and label before calculating

A quick sketch with known values, axis directions, and signs prevents most kinematics mistakes. In graph questions, annotate slopes and areas instead of guessing.

Know when equations are not enough

If acceleration is not constant, kinematic equations are not valid. In AP Physics 1, that often signals the need to interpret a graph, reason qualitatively, or use a different model introduced later in the course.

Why kinematics matters beyond this unit

Kinematics builds your ability to translate physical situations into mathematical relationships and back again. When you later study forces, you will connect acceleration to net force. When you study energy, you will relate speed to kinetic energy. When you study momentum, you will rely on careful vector thinking in multiple dimensions. Mastering kinematics is not just preparation for the AP exam, it is preparation for thinking like a physicist.