AP Physics 1: Kinematic Equations Derivation

Understanding the kinematic equations is not just about memorizing formulas for the exam; it is about grasping the fundamental language of motion. These equations allow you to predict an object's future position or velocity, a powerful tool applicable from designing roller coasters to programming robotics. This derivation connects the simple, intuitive ideas of speed and change in speed to the powerful mathematical tools you will use throughout physics.

From Definitions to Relationships: Velocity and Acceleration

All kinematics begins with two core definitions. First, average velocity ($v_{avg}$) is defined as the displacement divided by the time interval over which that displacement occurs: $v_{avg}=\frac{\Delta x}{\Delta t}$. Graphically, on a position-time plot, it is the slope of the secant line between two points. Second, acceleration ($a$) is the rate of change of velocity: $a=\frac{\Delta v}{\Delta t}$. For the standard kinematic equations to be valid, we make a critical assumption: acceleration is constant. This means the velocity changes at a steady, uniform rate, making its graph a straight line.

With constant acceleration, a crucial simplification occurs: the average velocity over an interval is simply the arithmetic mean of the initial and final velocities. This is only true when acceleration is uniform. Therefore, we can write: $$v_{avg}=\frac{v_i+v_f}{2}$$ Combining this with the definition $v_{avg}=\frac{\Delta x}{\Delta t}$, we arrive at our first useful kinematic relationship: $$\Delta x=\frac{1}{2}(v_i+v_f)t$$ Equation 1 is most useful when you know the initial and final velocities and the time of travel, but do not know and do not need the acceleration.

Deriving the Velocity-Time Relationship

Starting from the very definition of constant acceleration, $a=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t}$, we can solve algebraically for the final velocity. This gives us the simplest kinematic equation: $$v_f=v_i+at$$ Equation 2 defines a linear relationship between velocity and time. It is your direct tool for finding how fast an object is moving after a period of constant acceleration, or for finding the time required to reach a certain speed. Graphically, this is the equation of the line on a velocity-time graph, where $v_i$ is the y-intercept and $a$ is the slope.

Deriving the Position-Time Relationship

We can derive an equation that relates displacement directly to time, eliminating final velocity. Begin with Equation 1: $\Delta x=\frac{1}{2}(v_i+v_f)t$. Then, substitute $v_f$ from Equation 2 ($v_f=v_i+at$) into this expression: $$\Delta x=\frac{1}{2}(v_i+(v_i+at))t$$ Simplifying the expression inside the parentheses gives: $$\Delta x=\frac{1}{2}(2v_i+at)t$$ Which simplifies to our third key equation: $$\Delta x=v_{i}t+\frac{1}{2}a{t}^2$$ Equation 3 is quintessential for problems where you know the initial velocity, acceleration, and time, and you need to find the displacement (or vice-versa). The term $v_{i}t$ represents the displacement the object would have had if it continued at constant initial velocity, while $\frac{1}{2}a{t}^2$ represents the additional displacement due to the constant acceleration.

Deriving the Velocity-Position Relationship

Finally, we derive an equation that relates velocity to displacement, eliminating time. This is often the most algebraically involved derivation for students. Start again with Equation 1 and Equation 2:

1. $\Delta x=\frac{1}{2}(v_i+v_f)t$
2. $v_f=v_i+at$ (which can be rearranged to $t=\frac{v_f-v_i}{a}$)

Now, substitute the expression for $t$ from the rearranged Equation 2 into Equation 1: $$\Delta x=\frac{1}{2}(v_i+v_f)\left(\frac{v_f-v_i}{a}\right)$$ Notice that $(v_i+v_f)(v_f-v_i)$ is a difference of squares, simplifying to $v_f^2-v_i^2$. Making this substitution gives: $$\Delta x=\frac{1}{2a}(v_f^2-v_i^2)$$ Solving for $v_f^2$ yields the fourth standard kinematic equation: $$v_f^2=v_i^2+2a\Delta x$$ Equation 4 is incredibly powerful for problems where time is not given or is not requested. It allows you to directly connect a change in position to a change in the square of the velocity.

Common Pitfalls

1. Applying the equations to non-constant acceleration: The most fundamental error is using these four equations when acceleration is not constant. For example, the motion of a spring or a pendulum involves changing acceleration, so these kinematics equations are invalid. Always verify the "constant acceleration" condition is met.
2. Misinterpreting vector signs in one-dimensional motion: Kinematics is vector-based. You must define a positive direction (e.g., "up is positive") and consistently assign positive or negative signs to displacement, velocity, and acceleration. A common mistake is substituting a downward acceleration (e.g., -9.8 m/s²) into an equation while treating displacement as a positive magnitude, leading to incorrect answers.
3. Mixing and matching components in two dimensions: These are one-dimensional equations. For projectile motion, you must separate the motion into independent horizontal (where $a_x=0$) and vertical (where $a_y=-9.8{\text{ m/s}}^2$) components. A major pitfall is trying to use a single equation with mixed x and y variables. Instead, solve the horizontal and vertical motions separately, using time as the common linking variable.
4. Algebraic manipulation errors: Especially with Equation 4 ($v_f^2=v_i^2+2a\Delta x$), students often forget to take the square root at the end to solve for $v_f$, or make mistakes when solving for $\Delta x$ or $a$. When solving multi-step problems, work symbolically as long as possible before plugging in numbers. This makes it easier to track units and catch errors.

Summary

• The four standard kinematic equations are derived from the definitions of average velocity and constant acceleration, combined with the key realization that constant acceleration makes average velocity equal to $\frac{v_i+v_f}{2}$.
• Each equation has a "missing variable," which makes it particularly useful for solving problems where that variable is unknown or irrelevant: Eq. 1 misses $a$, Eq. 2 misses $\Delta x$, Eq. 3 misses $v_f$, and Eq. 4 misses $t$.
• The core assumption for all equations is constant acceleration. Their application to scenarios with changing acceleration will yield incorrect results.
• Successful problem-solving requires a disciplined, four-step approach: (1) identify knowns and the target unknown, (2) select the equation missing the irrelevant variable, (3) substitute known values with careful attention to vector signs, and (4) solve algebraically.
• For two-dimensional projectile motion, you must decompose the motion into independent horizontal and vertical components, applying the one-dimensional kinematic equations to each component separately.
• Mastering these derivations transforms the equations from a list of formulas to be memorized into a logical, interconnected toolkit for analyzing and predicting motion, a foundational skill for all subsequent physics study.