How to Calculate Distance as a Function of Time with Non-Uniform Acceleration

Calculus, specifically integration, is the key to understanding and calculating distance as a function of time when dealing with non-uniform acceleration. This article provides a step-by-step guide to solve this problem and highlights the fundamental principles and practical applications.

Understanding the Problem

When an object is moving with non-uniform acceleration, the acceleration is a function of time. The goal is to find the position (( s(t) )) as a function of time.

The Relationship Between Acceleration, Velocity, and Position

The fundamental relationships between acceleration, velocity, and position are given by:

Acceleration:

a t # d v t {dt}

Velocity:

v t # d s t {dt}

Integrating Acceleration to Find Velocity

To find the velocity (( v(t) )) as a function of time, integrate the acceleration function:

v t v 0 ∫ 0 t a t dt

Here, ( v_0 ) is the initial velocity at ( t 0 ).

Integrating Velocity to Find Position

Next, to find the position (( s(t) )) as a function of time, integrate the velocity function:

s t s 0 ∫ 0 t v t dt

Here, ( s_0 ) is the initial position at ( t 0 ).

An Example: Acceleration as ( at kt )

Suppose the acceleration is given by ( at kt ) where ( k ) is a constant.

Step 1: Find Velocity

Integrate the acceleration to find the velocity:

v t v 0 ∫ 0 t k t 1 dt

v t v 0 1 2 k t 2

Step 2: Find Position

Integrate the velocity to find the position:

s t s 0 ∫ 0 t ∫ 0 t v t dt

s t s 0 v 0 t 1 3 k t 3

Conclusion

This method provides a systematic approach to finding the distance as a function of time under non-uniform acceleration. Depending on the specific form of the acceleration function (( a(t) )), the integration steps may vary, but the overall process remains the same.

Integrating the acceleration twice, first to obtain velocity and then to obtain distance, is a fundamental concept in calculus and physics.