2.10 Non-uniform acceleration

 Page 1 / 5

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

$\frac{đ}{đx}{x}^{n}=n{x}^{n-1};\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\frac{đ}{đx}{\left(ax+b\right)}^{n}=na{\left(ax+b\right)}^{n-1}$ $\frac{đ}{đx}\mathrm{sin}ax=a\mathrm{cos}ax;\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\frac{đ}{đx}\mathrm{cos}ax=-a\mathrm{sin}ax$ $\frac{đ}{đx}{e}^{x}={e}^{x};\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\frac{đ}{đx}{\mathrm{log}}_{e}x=\frac{1}{x}$

Integration

$\int {x}^{n}đx=\frac{{x}^{n+1}}{n+1};\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\int {\left(ax+b\right)}^{n}đx=\frac{{\left(ax+b\right)}^{n+1}}{a\left(n+1\right)}$ $\int \mathrm{sin}axđx=-\frac{\mathrm{cos}ax}{a};\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\int \mathrm{cos}axđx=\frac{\mathrm{sin}ax}{a}$ $\int {e}^{x}đx={e}^{x};\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\int \frac{đx}{x}=\mathrm{log}{}_{e}x$

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 :

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

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 :

What are the system of units
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
58asagravitasnal firce
Amar
water boil at 100 and why
what is upper limit of speed
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
which colour has the shortest wavelength in the white light spectrum
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
x=5.8-3.22 x=2.58
what is the definition of resolution of forces
what is energy?
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
n=a+b/T² find the linear express
أوك
عباس
Quiklyyy
Moment of inertia of a bar in terms of perpendicular axis theorem
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?