Derive first equation of motion by graphical method.

The first equation of motion relates to the uniform acceleration of an object. To derive it using a graphical method, we can visualize the relationship between displacement, initial velocity, final velocity, and time.

Graphical Representation

Consider a velocity-time graph where:

• The x-axis represents time (t).
• The y-axis represents velocity (v).

Steps to Derive the Equation

1. **Initial Velocity (u)**: Start at point A on the y-axis, which indicates the initial velocity of the object.

2. **Final Velocity (v)**: After a time interval (t), the object reaches point B, representing the final velocity.

3. **Acceleration (a)**: The slope of the line connecting points A and B represents acceleration. It can be calculated as:

a = (v - u) / t

Finding Displacement

The area under the velocity-time graph gives the displacement (s). The shape formed is a trapezoid when considering initial and final velocities. The area can be calculated as:

s = (1/2) × (u + v) × t

Final Equation

By rearranging the acceleration formula and substituting it into the displacement equation, we arrive at:

v = u + at

This equation shows how the final velocity (v) depends on the initial velocity (u), acceleration (a), and time (t).