Given The Equation $s = U T + \frac{1}{2} A T^2$, Solve For $u$.

Mar 12, 2025 by ADMIN 69 views

**Solving for Initial Velocity in the Equation of Motion**

Introduction

The equation of motion, given by $s = u t + \frac{1}{2} a t^2$, is a fundamental concept in physics that describes the relationship between an object's displacement, initial velocity, acceleration, and time. In this article, we will focus on solving for the initial velocity, $u$, in the equation of motion. This is a crucial step in understanding the motion of objects and is essential in various fields such as engineering, physics, and mathematics.

Understanding the Equation of Motion

The equation of motion is a quadratic equation that relates an object's displacement, $s$, to its initial velocity, $u$, acceleration, $a$, and time, $t$. The equation is given by:

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

In this equation, $s$ represents the displacement of the object, $u$ is the initial velocity, $a$ is the acceleration, and $t$ is the time.

Solving for Initial Velocity

To solve for the initial velocity, $u$, we need to isolate $u$ on one side of the equation. We can do this by subtracting $\frac{1}{2} a t^2$ from both sides of the equation:

s - \frac{1}{2} a t^2 = u t

Next, we can divide both sides of the equation by $t$ to get:

\frac{s - \frac{1}{2} a t^2}{t} = u

Simplifying the left-hand side of the equation, we get:

u = \frac{s - \frac{1}{2} a t^2}{t}

Derivation of the Formula

To derive the formula for solving for initial velocity, we can start with the equation of motion:

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

We can rearrange this equation to get:

u t = s - \frac{1}{2} a t^2

Next, we can divide both sides of the equation by $t$ to get:

u = \frac{s - \frac{1}{2} a t^2}{t}

This is the formula for solving for initial velocity, $u$.

Example Problem

Suppose we have an object that moves with an initial velocity of $u$, an acceleration of $a$, and a displacement of $s$ after a time of $t$. If the displacement is $s = 10$ m, the acceleration is $a = 2$ m/s$^2$, and the time is $t = 5$ s, what is the initial velocity, $u$?

Using the formula for solving for initial velocity, we get:

u = \frac{s - \frac{1}{2} a t^2}{t}

Substituting the values, we get:

u = \frac{10 - \frac{1}{2} (2) (5)^2}{5}

Simplifying the expression, we get:

u = \frac{10 - 25}{5}

u = \frac{-15}{5}

In this article, we have derived the formula for solving for initial velocity, $u$, in the equation of motion. We have also provided an example problem to illustrate the application of the formula. The formula is given by: