IN THIS LESSON

Understand how to interpret, visualize, and quantify motion.

9/28/25: Welcome to the Mechanics section of the Physics course! This chapter will be based on the AP Physics C: Mechanics course or the typical introductory physics course at a university.

I figured out how to do the formatting a little better on this page than on the dot products page, so hopefully, this is more readable. I will still have the PDF version of this lesson attached at the bottom of the page.

Kinematics is the mathematical study of motion, independent of the forces that cause it. The fundamental quantities we will use are as follows:
  • Time (𝑑): The amount of time that has passed by a certain moment, typically in seconds.
  • Position (π‘₯(𝑑)): The location of an object relative to an arbitrary origin. Note: The origin can be placed wherever is most convenient for problem-solving.
  • Displacement (Ξ”π‘₯): The change in position, defined as Ξ”π‘₯ = π‘₯𝑓 βˆ’ π‘₯0. This is a vector quantity and is independent of the chosen origin.
  • Average velocity (𝑣ave): The change in position over time, defined as Ξ”π‘₯/Δ𝑑. This is also a vector, but it does not describe motion at a specific instant.
  • Average acceleration (π‘Žave): The change in velocity over time, defined as Δ𝑣/Δ𝑑. Another vector, describing how velocity changes over an interval of time.
  • Instantaneous velocity (𝑣(𝑑)): Velocity at a specific time, defined as the derivative of position with respect to time, 𝑑π‘₯/𝑑𝑑. Also a vector.
  • Instantaneous acceleration (π‘Ž(𝑑)): Acceleration at a specific time, defined as the derivative of velocity with respect to time, 𝑑𝑣/𝑑𝑑. Also a vector.

Note: A vector is a quantity that has both magnitude and direction, as opposed to a scalar quantity which only has magnitude. Direction in kinematics is often expressed as signs of quantities, i.e. -5 m/s and 5 m/s are velocities in opposite directions.

Graphical representation of instantaneous velocity. A derivative of a function at a point is just the slope of the function at that point.

We can represent motion visually with graphs:
  • The slope of an π‘₯ vs. 𝑑 graph gives the velocity. A positive slope indicates velocity in the positive direction.
  • The area under a 𝑣 vs. 𝑑 graph gives the displacement.

Graphical representation of displacement. The area under a velocity graph represents the displacement from t = 0 until the current time t. The area under a graph is also known as an integral.

From calculus and algebra, we derive a set of standard kinematic equations of motion for motion of constant acceleration:
    1. Ξ”π‘₯ = 𝑣0Δ𝑑 + Β½π‘ŽΞ”π‘‘2
    2. Ξ”π‘₯ = Β½(𝑣0 + 𝑣𝑓)Δ𝑑
    3. Ξ”π‘₯ = 𝑣𝑓Δ𝑑 + Β½π‘ŽΞ”π‘‘2
    4. 𝑣𝑓 = 𝑣0 + π‘ŽΞ”π‘‘
    5. 𝑣𝑓2 = 𝑣02 + 2π‘ŽΞ”π‘₯
Which equation you use depends on the variables you are given, the one you are solving for, and which variable can be ignored (the "don't know don't care" or β€œdk/dc variable”).

Table showing which equations should be used based on which variable is your β€œdk/dc variable.” The variables you were given values for and the variable you are looking for should all be checked in the table.

When solving kinematics problems, it is always best to start by sketching the motion. Label all known values, the unknown variable you are solving for, and the variable you don’t care about (dk/dc). This helps clarify which kinematic equation is most appropriate.

One important application of kinematics is projectile motion, which is based on these tenants:
  • A projectile is any object acted on only by gravity.
  • Motion in the π‘₯ and 𝑦 directions are independent from each other. This means you should split the displacement and velocity vectors into horizontal and vertical components (Ξ”π‘₯, Δ𝑦, 𝑣π‘₯0, 𝑣𝑦0).
  • Acceleration in the horizontal direction is always zero, while in the vertical direction it is always 𝑔 = 9.81 m/s2 in the downward direction.

The two most important equations in projectile motion are as follows:
    1. Projectile Equation: Δ𝑦 = Ξ”π‘₯tanπœƒ - 𝑔Δπ‘₯2/2𝑣02cos2πœƒ
    2. Range Equation: Ξ”π‘₯ = 𝑣02sin(2πœƒ)/𝑔

Diagram representing projectile motion and its relevant variables. The only acceleration acting on a projectile is gravitation acceleration downwards as mentioned above.

These equations show that:
  • Vertical motion is symmetrical.
  • Horizontal components of velocity are unaffected by gravity.

Finally, kinematics relies on reference frames:
  • Inertial reference frames move at constant velocity, and Newton’s First Law (otherwise known as the Law of Inertia) applies only in these frames. For our purposes, we will only work within inertial reference frames.
  • You can define any reference frame you want, as long as the axes are perpendicular and right-handed, meaning the positive 𝑦-axis is 90Β° counterclockwise from the positive π‘₯-axis.
  • Choosing a convenient reference frame often simplifies the problem significantly.
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