Understanding the Derivation and Application of S ut 1/2at^2

Understanding the principles of motion in physics is fundamental for anyone studying dynamics. One of the key equations used in kinematics is the equation for distance traveled:

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

Here, S represents the distance, u is the initial velocity, a is the acceleration, and t is the time. This equation is essential for calculating the distance traveled by an object under constant acceleration. It is often questioned why the equation isn't simplified to S utat^2, which would be incorrect.

Derivation of the Equation

Let's delve into the derivation of this equation using the principles of calculus and Newton's laws of motion.

Nature of Acceleration

According to Newton's equations of motion, we know that:

[ v u at ]

However, it is crucial to recognize that the velocity changes instantaneously as the object is under constant acceleration. Therefore, the distance covered in a small time interval (dt) is represented as:

[ ds v , dt ]

Now, let's substitute (v u at) into the equation:

[ ds (u at) , dt ]

To find the total distance (s) covered over a time interval (t), we integrate both sides:

[ int_{0}^{s} ds int_{0}^{t} (u at) , dt ]

This yields:

[ s int_{0}^{t} u , dt int_{0}^{t} at , dt ]

Evaluating these integrals:

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

Understanding Velocity and Distance

It's important to point out that (v s/t) gives the average velocity, not the final velocity given by the equation (v u at). This distinction is crucial for applying the correct equations in different situations. For instance, if an object is moving with constant acceleration, the distance traveled is best calculated using:

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

This equation accurately represents the distance covered by an object under constant acceleration at any given time.

Alternative Formulations

There are other formulations of the equations of motion in kinematics, such as:

[ v^2 u^2 2as ]

Substituting (v u at) into this equation, we get:

[ (u at)^2 u^2 2as ]

Expanding and simplifying:

[ u^2 2uat a^2 t^2 u^2 2as ]

Subtracting (u^2) and (2as) from both sides:

[ a^2 t^2 - 2uat 0 ]

Dividing by (2a):

[ 2as a^2 t^2 - 2ut ]

Thus:

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

This confirms the initial equation and demonstrates why the (1/2) term is necessary for the correct expression of distance covered under constant acceleration.

Conclusion

Remember, in the absence of acceleration, (v s/t) provides an accurate measure of average velocity. However, when dealing with constant acceleration, the correct formula is (S ut frac{1}{2}at^2). Understanding these principles is essential for applying the right formulas in various scenarios.

In summary, the key takeaway is that (S ut frac{1}{2}at^2) is the correct expression for distance traveled under constant acceleration, and it is derived from Newton's laws and calculus.