Equations of Motion - Part I

We are now familiar with terms such as speed, velocity, acceleration, force, motion, etc. We have also studied Newtonís laws of motion. We know what are scalars and vectors. Now we will see how we can predict motions or positions of particles or bodies given a few initial parameters.

The equations of motion that we will study in this chapter will equip us adequately to make predictions of motions of particles or bodies at times away from the initial parameters. For example, if you are in a car that is starting from zero velocity, and if we are given the acceleration of the car, you will know where the car will be after say five minutes. Or if you throw a stone with a certain force vertically upwards, you will be able to tell the time that will be taken for the stone to return back. 

There are basically three equations of motion. They are valid for translation motion of a body only. In a translational motion, when a body travels from position 1 to position 2, all particles in the body undergo identical motion parallel to each other and at the same time. This means that the displacement of each and every particle of the body is identical. Translational motion in a straight line is called a rectilinear motion. Other types of motion like rotational motion have complicated angular velocity and angular accelerations involved with it.  For the present study, we consider the equations of motion for translational motion only.

What we will study in this chapter :

1. Revision of terms such as speed, velocity, acceleration
2. First equation of motion  
3. Second equation of motion
4. Third equation of motion

1. Revision of terms such as speed, velocity, acceleration
To recapitulate a few definitions, studied in the last few chapters, remember that

Speed = Distance   
                time

Velocity of a body is the distance it travels in a particular direction in a given time. Velocity and speed have the same definitions, but velocity is a vector quantity, whereas speed is a scalar quantity.

                                                     
 v  =  s , distance = s, time = t. The MKS unit of   v  is m/s
         t

Uniform acceleration occurs when a body undergoes same incremental increase in velocity in equal intervals of time. Non uniform motion occurs when the increase in velocity is not constant in equal intervals of time. For the present study we will consider motion under uniform acceleration only.

                                                    
 a   =   v ,  velocity =  v , time = t. The MKS unit of  a  is m/s²  
           t

 a  =    s
           t²

2. First equation of motion  
The first equation of motion follows directly from the definition of acceleration.

Consider a body to be stationary at time t = 0

     
Let    u    be the initial  velocity of the  body.

The body stops after a time = t sec.

                                                    
When the body is stopping, its final velocity is v .

                                             (final velocity ñ initial velocity)
Thus acceleration     a   = 
                                          time taken for the change of velocity

            
 a    =    (  v   -  u  )
                    t

For writing equations, we can ignore the vector signs.

Thus   at = (v - u)

           v = u + at        

This is the first equation of motion. It is used to calculate the velocity achieved by a body after time t, if the initial velocity and uniform acceleration are known.

Example 1:  A car starts from rest and attains a velocity of 60km/hour and stops after 10 minutes. What was the acceleration of the car ?

                          
In this example, u  =  0,    v  =  60 km/hour = 60 km/60 minute

                                                                 = 1 km/minute = 1000m/60s = 16.67 m/s

Substituting this value in the equation  v = u + at, and t = 10 minutes = 600s
We get a =0.027 m/s².

Example 2: A car  is moving at 10m/s. Brakes are applied and the car comes to rest in 20 seconds. Calculate the deceleration of the car.

Deceleration is opposite of acceleration or negative acceleration.

In this problem v = 0, t = 20 s, u = 10m/s

In the equation v = u +at we get u = -at,

Thus a = - 0.5 m/ s².

The negative sign indicates deceleration.  

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