# Equations of Uniformly Accelerated Motion The equations of motion are a set of fundamental formulas in kinematics that describe the relationship between displacement, velocity, acceleration, and time for an object moving with **constant acceleration** in a straight line. These equations allow us to predict the state of an object's motion at any given time, provided the acceleration is uniform. The five variables involved are: * **s**: displacement * **vᵢ**: initial velocity * **vƒ**: final velocity * **a**: constant acceleration * **t**: time interval ## Derivation of the Equations The three main equations of motion can be derived graphically from a **velocity-time (v-t) graph**. For an object with uniform acceleration, the v-t graph is a straight line. * The **slope** of the v-t graph represents **acceleration**. * The **area under** the v-t graph represents **displacement**. Consider an object that accelerates from an initial velocity $v_i$ to a final velocity $v_f$ over a time interval $t$. ### 1. First Equation of Motion: Velocity-Time Relation This equation relates the final velocity of an object to its initial velocity, acceleration, and the time elapsed. **Formula**: $$ v_f = v_i + at $$ **Derivation**: Acceleration (*a*) is defined as the slope of the velocity-time graph. $$ \text{Slope} = a = \frac{\text{Change in Velocity}}{\text{Change in Time}} = \frac{v_f - v_i}{t} $$ Rearranging this equation to solve for $v_f$, we get: $$ at = v_f - v_i $$ $$ v_f = v_i + at $$ ### 2. Second Equation of Motion: Displacement-Time Relation This equation calculates the displacement of an object given its initial velocity, acceleration, and the time interval. **Formula**: $$ s = v_i t + \frac{1}{2} a t^2 $$ **Derivation**: Displacement (*s*) is the total area under the v-t graph. This area can be seen as a trapezium. The area of a trapezium is given by: $$ s = \text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times (\text{height}) $$ In our graph, the parallel sides are the initial velocity ($v_i$) and final velocity ($v_f$), and the height is time (*t*). $$ s = \frac{1}{2}(v_i + v_f)t $$ Now, we can substitute the expression for $v_f$ from the first equation of motion ($v_f = v_i + at$) into this area formula: $$ s = \frac{1}{2}(v_i + (v_i + at))t $$ $$ s = \frac{1}{2}(2v_i + at)t $$ Distributing the *t* gives the final form: $$ s = v_i t + \frac{1}{2} a t^2 $$ ### 3. Third Equation of Motion: Velocity-Displacement Relation This equation relates the final and initial velocities to acceleration and displacement, notably without requiring time. **Formula**: $$ 2as = v_f^2 - v_i^2 $$ **Derivation**: We start again with the formula for displacement as the area under the graph: $$ s = \frac{1}{2}(v_i + v_f)t $$ From the first equation of motion, we can express time as $t = \frac{v_f - v_i}{a}$. Substituting this expression for *t* into the displacement formula: $$ s = \frac{1}{2}(v_f + v_i)\left(\frac{v_f - v_i}{a}\right) $$ Multiplying the terms in the parentheses gives a difference of squares: $$ s = \frac{v_f^2 - v_i^2}{2a} $$ Rearranging this gives the final form: $$ 2as = v_f^2 - v_i^2 $$ To ensure these equations are physically valid, they must satisfy the **Principle of Homogeneity**, meaning the on both sides of the equation must be identical. ## Summary | **Equation Number** | **Formula** | **Variables Related** | **Variable Not Included** | | :--- | :--- | :--- | :--- | | **1st Equation** | $v_f = v_i + at$ | Velocity, Acceleration, Time | Displacement (s) | | **2nd Equation** | $s = v_i t + \frac{1}{2} a t^2$ | Displacement, Velocity, Time | Final Velocity (vƒ) | | **3rd Equation** | $2as = v_f^2 - v_i^2$ | Velocity, Acceleration, Displacement | Time (t) | These equations form the bedrock of kinematics and are essential for solving problems involving objects moving with constant acceleration, from a falling apple to a launching rocket. ## References - [Equations of Uniformly Accelerated Motion (FBISE Physics)](https://www.youtube.com/watch?v=RgvGsAgW83U)