17.5 Equations of motion (Page 22/3)

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Equations of motion

In this chapter we will look at the third way to describe motion. We have looked at describing motion in terms of graphs and words. In this section we examine equations that can be used to describe motion.

This section is about solving problems relating to uniformly accelerated motion. In other words, motion at constant acceleration.

The following are the variables that will be used in this section:

\begin{matrix} v_{i} & = & initial velocity (m \cdot s^{- 1}) at t = 0 s \\ v_{f} & = & final velocity (m \cdot s^{- 1}) at time t \\ Δ x & = & displacement (m) \\ t & = & time (s) \\ Δ t & = & time interval (s) \\ a & = & acceleration (m \cdot s^{- 1}) \end{matrix}

v_{f} = v_{i} + a t

Δ x = \frac{(v_{i} + v_{f})}{2} t

Δ x = v_{i} t + \frac{1}{2} a t^{2}

v_{f}^{2} = v_{i}^{2} + 2 a Δ x

The questions can vary a lot, but the following method for answering them will always work. Use this when attempting a question that involves motion with constant acceleration. You need any three known quantities ( $v_{i}$ , $v_{f}$ , $Δ x$ , $t$ or $a$ ) to be able to calculate the fourth one.

Read the question carefully to identify the quantities that are given. Write them down.
Identify the equation to use. Write it down!!!
Ensure that all the values are in the correct unit and fill them in your equation.
Calculate the answer and fill in its unit.

Interesting fact

Galileo Galilei of Pisa, Italy, was the first to determined the correct mathematical law foracceleration: the total distance covered, starting from rest, is proportional to the square of the time. He also concluded thatobjects retain their velocity unless a force – often friction – acts upon them, refuting the accepted Aristotelian hypothesis thatobjects "naturally" slow down and stop unless a force acts upon them. This principle was incorporated into Newton's laws of motion(1st law).

Finding the equations of motion

The following does not form part of the syllabus and can be considered additional information.

Derivation of [link]

According to the definition of acceleration:

a = \frac{Δ v}{t}

where $Δ v$ is the change in velocity, i.e. $Δ v = v_{f}$ - $v_{i}$ . Thus we have

\begin{matrix} a & = & \frac{v_{f} - v_{i}}{t} \\ v_{f} & = & v_{i} + a t \end{matrix}

Derivation of [link]

We have seen that displacement can be calculated from the area under a velocity vs. time graph. For uniformly accelerated motion the most complicated velocity vs. time graph we can have is a straight line. Look at the graph below - it represents an object with a starting velocity of $v_{i}$ , accelerating to a final velocity $v_{f}$ over a total time $t$ .

To calculate the final displacement we must calculate the area under the graph - this is just the area of the rectangle added to the area of the triangle. This portion of the graph has been shaded for clarity.

\begin{matrix} {Area}_{▵} & = & \frac{1}{2} b \times h \\ = & \frac{1}{2} t \times (v_{f} - v_{i}) \\ = & \frac{1}{2} v_{f} t - \frac{1}{2} v_{i} t \end{matrix}

\begin{matrix} {Area}_{□} & = & ℓ \times b \\ = & t \times v_{i} \\ = & v_{i} t \end{matrix}

\begin{matrix} Displacement & = & {Area}_{□} + {Area}_{▵} \\ Δ x & = & v_{i} t + \frac{1}{2} v_{f} t - \frac{1}{2} v_{i} t \\ Δ x & = & \frac{(v_{i} + v_{f})}{2} t \end{matrix}

Derivation of [link]

This equation is simply derived by eliminating the final velocity $v_{f}$ in [link] . Remembering from [link] that

v_{f} = v_{i} + a t

then [link] becomes

\begin{matrix} Δ x & = & \frac{v_{i} + v_{i} + a t}{2} t \\ = & \frac{2 v_{i} t + a t^{2}}{2} \\ Δ x & = & v_{i} t + \frac{1}{2} a t^{2} \end{matrix}

Derivation of [link]

This equation is just derived by eliminating the time variable in the above equation. From [link] we know

t = \frac{v_{f} - v_{i}}{a}

Substituting this into [link] gives

\begin{matrix} Δ x & = & v_{i} (\frac{v_{f} - v_{i}}{a}) + \frac{1}{2} a {(\frac{v_{f} - v_{i}}{a})}^{2} \\ = & \frac{v_{i} v_{f}}{a} - \frac{v_{i}^{2}}{a} + \frac{1}{2} a (\frac{v_{f}^{2} - 2 v_{i} v_{f} + v_{i}^{2}}{a^{2}}) \\ = & \frac{v_{i} v_{f}}{a} - \frac{v_{i}^{2}}{a} + \frac{v_{f}^{2}}{2 a} - \frac{v_{i} v_{f}}{a} + \frac{v_{i}^{2}}{2 a} \\ 2 a Δ x & = & - 2 v_{i}^{2} + v_{f}^{2} + v_{i}^{2} \\ v_{f}^{2} & = & v_{i}^{2} + 2 a Δ x \end{matrix}

This gives us the final velocity in terms of the initial velocity, acceleration and displacement and is independent of the time variable.

Questions & Answers

if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand

Syamthanda Reply

hey , can you please explain oxidation reaction & redox ?

Boitumelo Reply

hey , can you please explain oxidation reaction and redox ?

Boitumelo

for grade 12 or grade 11?

Sibulele

the value of V1 and V2

Tumelo Reply

advantages of electrons in a circuit

Rethabile Reply

we're do you find electromagnetism past papers

Ntombifuthi

what a normal force

Tholulwazi Reply

it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface

Sihle

what is physics?

Petrus Reply

what is the half reaction of Potassium and chlorine

Anna Reply

how to calculate coefficient of static friction

Lisa Reply

how to calculate static friction

Lisa

How to calculate a current

Tumelo

how to calculate the magnitude of horizontal component of the applied force

Mogano

How to calculate force

Monambi

a structure of a thermocouple used to measure inner temperature

Anna Reply

a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4

Amahle Reply

How is energy being used in bonding?

Raymond Reply

what is acceleration

Syamthanda Reply

a rate of change in velocity of an object whith respect to time

Khuthadzo

how can we find the moment of torque of a circular object

Kidist

Acceleration is a rate of change in velocity.

Justice

t =r×f

Khuthadzo

how to calculate tension by substitution

Precious Reply

Shongi

Leago

use fnet method. how many obects are being calculated ?

Khuthadzo

khuthadzo hii

Hulisani

how to calculate acceleration and tension force

Lungile Reply

you use Fnet equals ma , newtoms second law formula

Masego

please help me with vectors in two dimensions

Mulaudzi Reply

how to calculate normal force

Mulaudzi

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Source: OpenStax, Siyavula textbooks: grade 10 physical science [caps]. OpenStax CNX. Sep 30, 2011 Download for free at http://cnx.org/content/col11305/1.7

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