2.10 Non-uniform acceleration

Non-uniform acceleration constitutes the most general description of motion. It refers to variation in the rate of change in velocity. Simply put, it means that acceleration changes during motion. This variation can be expressed either in terms of position (x) or time (t). We understand that if we can describe non-uniform acceleration in one dimension, we can easily extend the analysis to two or three dimensions using composition of motions in component directions. For this reason, we shall confine ourselves to the consideration of non-uniform i.e. variable acceleration in one dimension.

In this module, we shall describe non-uniform acceleration using expressions of velocity or acceleration in terms of either of time, “t”, or position, “x”. We shall also consider description of non-uniform acceleration by expressing acceleration in terms of velocity. As a matter of fact, there can be various possibilities. Besides, non-uniform acceleration may involve interpretation acceleration - time or velocity - time graphs.

Accordingly, analysis of non-uniform acceleration motion is carried out in two ways :

• Using calculus
• Using graphs

Analysis using calculus is generic and accurate, but is limited to the availability of expression of velocity and acceleration. It is not always possible to obtain an expression of motional attributes in terms of “x” or “t”. On the other hand, graphical method lacks accuracy, but this method can be used with precision if the graphs are composed of regular shapes.

Using calculus involves differentiation and integration. The integration allows us to evaluate expression of acceleration for velocity and evaluate expression of velocity for displacement. Similarly, differentiation allows us to evaluate expression of position for velocity and evaluate expression of velocity for acceleration. We have already worked with expression of position in time. We shall work here with other expressions. Clearly, we need to know a bit about differentiation and integration before we proceed to analyze non-uniform motion.

Important calculus results

Integration is anti-differentiation i.e. an inverse process. We can compare differentiation and integration of basic algebraic, trigonometric, exponential and logarithmic functions to understand the inverse relation between processes. In the next section, we list few important differentiation and integration results for reference.

Differentiation

$$\frac{đ}{đx}{x}^n=n{x}^{n-1};\frac{đ}{đx}{\left(ax+b\right)}^n=na{\left(ax+b\right)}^{n-1}$$ $$\frac{đ}{đx}\sin ax=a\cos ax;\frac{đ}{đx}\cos ax=-a\sin ax$$ $$\frac{đ}{đx}{e}^x=e^x;\frac{đ}{đx}{\log }_{e}x=\frac{1}{x}$$

Integration

$$\int x^nđx=\frac{x^{n+1}}{n+1};\int {\left(ax+b\right)}^nđx=\frac{{\left(ax+b\right)}^{n+1}}{a\left(n+1\right)}$$ $$\int \sin axđx=-\frac{\cos ax}{a};\int \cos axđx=\frac{\sin ax}{a}$$ $$\int e^xđx=e^x;\int \frac{đx}{x}=\log {}_{e}x$$

Velocity and acceleration is expressed in terms of time “t

Let the expression of acceleration in x is given as function a(t). Now, acceleration is related to velocity as :

$$a\left(t\right)=\frac{đv}{đt}$$

We obtain expression for velocity by rearranging and integrating :

$$\Rightarrow đv=a\left(t\right)đt$$ $$\Rightarrow \text{Δ}v=\int a\left(t\right)đt$$

This relation yields an expression of velocity in "t" after using initial conditions of motion. We obtain expression for position/ displacement by using defining equation, rearranging and integrating :