Kinematics Graphical Derivation — Study Pack Summary & Study Notes

📚 Overview

This section summarizes how to derive the standard equations of motion using graphical methods (velocity–time graphs). The graphical approach uses two key ideas: the slope of a velocity–time graph gives acceleration, and the area under a velocity–time graph gives displacement.

📈 Velocity–Time Graphs

A velocity–time graph has velocity on the vertical axis and time on the horizontal axis. For constant acceleration the graph is a straight line whose slope is the acceleration. For constant velocity the line is horizontal. For an object at rest the velocity line is the horizontal line at zero.

🧭 Key graphical relationships

• Slope = rise / run = change in velocity divided by change in time = acceleration.
• Area under curve = displacement (because velocity × time = displacement for each small interval).

✏️ Deriving the equations of motion (step-by-step)

1. From slope: if initial velocity is u, final velocity is v, time interval is t, then the slope gives

$v - u = a t$,

so rearranged: $v = u + a t$. This is the first standard equation.

1. From area (displacement as area under the velocity–time graph): split the trapezoidal area into a rectangle plus a triangle. The rectangle area = $u t$. The triangle area = $\frac{1}{2}(v - u)t$. Substitute $v - u = a t$ to get

$s = u t + \frac{1}{2} a t^2$.

1. Area as a trapezoid (average of parallel sides times height) gives

$s = \frac{u + v}{2} t$,

which emphasizes that for constant acceleration, average velocity = $\frac{u+v}{2}$.

1. Eliminating time using $t = \frac{v - u}{a}$ and substituting into the trapezoid form leads to

$v^2 = u^2 + 2 a s$.

1. A fifth useful rearrangement is obtained by solving the second equation for $s$ for problems where $v$ is not part of the data:

$s = v t - \frac{1}{2} a t^2$.

✅ Summary of standard forms (for constant acceleration)

• $v = u + a t$ (relates velocity, acceleration, time)
• $s = u t + \frac{1}{2} a t^2$ (displacement with initial velocity)
• $s = \frac{u + v}{2} t$ (displacement via average velocity)
• $v^2 = u^2 + 2 a s$ (velocity–displacement relation)
• $s = v t - \frac{1}{2} a t^2$ (alternate displacement form)

🔎 Tips and common pitfalls

• Always identify which variables are known and which are unknown; choose the equation that omits the unknown variable.
• Check sign conventions: upward/rightward positive by default; negative slopes mean negative acceleration.
• Area under the curve can be split into geometric shapes (rectangles, triangles, trapezoids) to compute displacement.

🧠 Quick conceptual checks

• If the velocity–time graph is horizontal at $v = 5\ \text{m/s}$, acceleration = 0 and displacement over time $t$ is $5t$.
• If the velocity–time graph is a line from $u$ to $v$ over time $t$, average velocity is $\frac{u+v}{2}$ and displacement = average velocity × $t$.

🔐 Google Sign-in page — Short notes

This page presents the Google Account sign-in interface. It prompts for an account email or phone, offers Forgot email? and Guest mode options, and provides links for Create account, language selection, and privacy/terms information.

⚖️ Privacy & modes

• Guest mode allows private browsing on a shared device without signing in. The page also links to Privacy and Terms documents that describe data handling when signing in.

📝 Relevance

While not a physics resource, the sign-in page is a commonly encountered web interface and reminds users to be mindful of account privacy when accessing online learning materials.