Deriving the Kinematics Equation S ut 1/2 at2

Understanding the Basics of Kinematics:

When dealing with motion under uniform acceleration, a fundamental equation emerges: S ut 1/2 at2. This equation describes the displacement S of an object starting with an initial velocity u and subjected to a constant acceleration a over a time period t. Letrsquo;s delve into a detailed derivation of this equation.

Definitions

S: Displacement of the object u: Initial velocity of the object a: Acceleration of the object t: Time period

Derivation of the Equation

Step 1: Understanding the Basic Definitions

The average velocity vavg can be defined as the mean of the initial and final velocities. The final velocity v after time t under uniform acceleration is given by:

[v u at]

Thus, the average velocity is:

[v_{avg} frac{u v}{2} frac{u (u at)}{2} u frac{1}{2}at]

Step 2: Relating Final Velocity to Acceleration

From the definition of acceleration:

[a frac{v - u}{t} Rightarrow v u at]

Step 3: Substituting Final Velocity into Average Velocity

Now, substituting the expression for v from the acceleration equation into the average velocity equation:

[v_{avg} frac{u u at}{2} frac{2u at}{2} u frac{1}{2}at]

Step 4: Calculating Displacement

The displacement S can be calculated using the average velocity over time:

[S v_{avg} cdot t]

Substituting the expression for vavg:

[S left(u frac{1}{2}atright) cdot t ut frac{1}{2}at^2]

Conclusion

Thus, the final equation representing the displacement of an object under uniform acceleration is:

[S ut frac{1}{2}at^2]

This equation is fundamental in kinematics and is widely used in physics to analyze motion.

Graphical Representation

To visualize the derivation, consider the following steps:

Step 1: Acceleration vs. Time

Graphing a vs. t is a simple, flat line at a constant value. This is a VERY BORING graph, but essential for understanding.

Step 2: Velocity vs. Time

To calculate the new value of v at each value of t, add the area under the a vs. t graph for that time step. This results in a straight line passing through the initial velocity u at t 0.

Step 3: Displacement vs. Time

Similarly, for s, repeat the process. Without a calculator, you can manually count squares on graph paper. The resultant shape is a parabola:

[S ut frac{1}{2}at^2]

A geometric proof involves plotting a point with the equation for directrix and observing the locus of points equidistant from both, confirming the parabolic shape.

Calculus Proof

Using calculus, the derivation follows:

Step 1: Definition of acceleration

[a frac{dv}{dt}]

Integrating:

[v u at]

Step 2: Definition of velocity in terms of displacement

[v frac{ds}{dt}]

Substituting the expression for v:

[ds (u at) dt]

Integrating:

[int ds int (u at) dt]

[s ut frac{1}{2}at^2 C]

Step 3: Applying Initial Condition

Given s 0 when t 0, we find:

[0 u(0) frac{1}{2}a(0)^2 C Rightarrow C 0]

Thus, the complete equation is:

[S ut frac{1}{2}at^2]

QED (Quod Erat Demonstrandum)

Conclusion

The equation S ut 1/2 at2 is a cornerstone in kinematics, representing how the displacement of an object changes over time under constant acceleration. This derivation and its graphical representations provide a comprehensive understanding of this fundamental principle in physics.