equations of motion

Equations of motion in physics

Parameters and equations of motion

The parameters of motion in straight lines are: displacement, speed, or velocity, acceleration and time
ALL the parameter is vector quantity except speed and time
Displacement: it is the distance travel in specific direction
Speed: Is the rate of change of distance with time. That is

v=\frac { s }{ t }

where v= speed S= distance and T = time
Uniform speed: a speed is said to be uniform if the rate of change of distance with time is constant
That is v=\frac { s }{ t } = constant 
Velocity: it is a speed measured in a given direction or the rate of change of displacement with time
NOTE: always remember that, speed and velocity are not the same in science. Velocity is a vector quantity, while speed is a scalar quantity
Acceleration: is the rate of change of velocity with time. That is =\frac { v }{ t }
Uniform acceleration: acceleration is said to be uniform, if the rate of change of velocity with time is constant.

That is =\frac { v }{ t }   =  constant

NOTE – if the velocity of a body is increasing with time, it is said to be accelerating, but if the velocity of the body is decreasing with time, the body is said to be experiencing retardation or deceleration. In this case acceleration is said to be negative.

The equation of motion

If a body starts with initial velocity u, accelerate uniformly along a straight line with acceleration a, and covers a distance S, in a time when its velocity reaches a final value v, then the distance covered is given by s=average velocity
s=\frac { v+u }{ 2 } \times t ……(1)
Also, by definition, acceleration a= rate of change of velocity, that is
a=\frac { v-u }{ t }
v=u+at ….(2)
Eliminating t from equation ( 1) and ( 2) we have
v^2 =u^2 +2as ….(3)
Eliminating v from equation ( 1) and( 2) we have
s=ut+\frac { 1 }{ 2 } at{ }^{ 2 }

So these are the four equations of motion

 s=\frac { v+u }{ 2 } \times t   ..... (1)  
  V = u + at ….(2)
 V^2 = u^2+ 2aS ........ (3)   
 s=ut+\frac { 1 }{ 2 } at{ }^{ 2 } ......(4)  

How to use the four equations of motion

Ensure that all the unit match, that is v in ms-1, S in m, a in  ms2 , t in s, or v in  kmh-  ,s in km, a in  kmh- 2 , and t in hours.
Each of the equations contains four of the five variables, u,v,s,a and t,and you are normally given the value of the three of them, to find one or two of the unknowns.
The best way to select which equation to use is to look at the problem and find out which of the five variables is not given, and then find the equation which does not contain the variables that is not given, and that gives you the required equation to use in solving the problem.

Example
A car moves from rest with an acceleration of 0.2ms find its velocity when it has move a distance of 50m.
Answer
Using,v^2 = u^2 + 2aS where v =?, u =0, a =0.2ms-2 and S = 50m
V2 = 02 = 0.2 x 50 = √20ms-2      

A car has uniform velocity of 108kmh^-1. How far does it travels in 30seconds.
Answer: since the car is moving with uniform velocity, then final velocity is equal to initial velocity, ie v = u
Using s=\frac { v+u }{ 2 } \times t
s=\frac { 2v }{ 2 } \times t=vt
But 108km/h =30m/s
S= vt = 30 \time 30 = 900m = 0.9km

Motion under gravity

Anybody falling under the influence of gravity alone falls with uniform acceleration. note that if two bodies are dropped from the same height at the same time, if they are under the influence of force of gravity alone, land at the same time.
The two object land at the same time, because, the inertia of an object or its resistance to acceleration, depends on its mass, which also determines its weight. Therefore large mass has large inertia also the value of acceleration under gravity is 9.8m/s^2 or 10m/s2^2
If the motion of a body is downward it has positive acceleration a+ but has a negative acceleration a-, if the motion is upward
NOTE this:
1 if a body is dropped from a height, the initial velocity is u= 0
2.when it is projected vertically upward the final velocity v=o.
3.when the body falls to the ground again the height above the ground is zero, s=o.

Body in circular motion

A body performing circular motion do that with uniform acceleration towards the centre of the circle, so a body under circular motion, is performing uniform circular motion .The two important quantities about circular motion they are period and frequency period .
Period : it is time taken to complete one circular motion and frequency is the number of circular motion completed in one second. Frequency is measured in Hz

Centrifuge

It is the device that is used to produce large force by whirling them in a circle.
Cause of motion
The only cause of motion of body is force. There are two types of forces.
Contact force: this type of force must be constant with body they act on. Example push or pull force. Tension , reaction and frictional forces
Field force: These forces do not require contact with the body before it act on the body. Example, gravitational force, electrical [electrostatic] force and magnetic force.

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