Equations of Motion - Part II

3. Second equation of motion  
The second equation of motion is derived from the definition of average velocity.    
                                 
If   u   is the initial velocity and    v   is the final velocity, then the average velocity can be defined as :

                             
Average velocity   =  (  v  +  u  ) / 2

The distance traveled  s  = average velocity  x time

Thus  s   =   ( v + u )   x t  
                       2 

Substituting the value of v from the first equation of motion,

we get s  =  (u + at + u)   x t  
                          2

Thus    s = ut + ‡ at²

This is the second equation of motion. This equation is useful in calculating the distance traveled by a body if the initial velocity, acceleration and time are known.

Example 1: A bullet covers a distance of 120 m in 10 seconds.  Find its acceleration at the end of 10 seconds.

Here u = 0, s = 120 m, t= 10 s

s = ‡ at²

Substituting the values of s and t, we get  a = 2.4 m/s².

Thus acceleration of the bullet after the end of 10 sec is 2.4 m/s².

Example 2: A ball is released from the top of a tower. It reaches the ground in 2 seconds. Find the height of the tower. The ball falls under the influence of gravitational acceleration of 9.8m/ s².

Here u = 0, t = 2 s,  a = 9.8 m/ s²

s = ‡ at²

Substituting the values of a and t, we get s = 39.2 m.

Since the acceleration due to gravity is always in the downward direction, we can take the value of a to be negative, This will give the result of s to be negative!! This looks a bit illogical. But if you consider it carefully, the equation shows that the ball is falling down towards the earth.

4. Third equation of motion
The third equation of motion follows from the first two equation of motion.

v = u + at

s = ut + ‡ at²

If we do not know t, we can eliminate t by substitution.

t = (v - u) / a

s =  u    x  (v - u)   +   1 a  x  (v - u) ²   
                          a              2                a²

After doing a few adjustments we get

        v²   = u² +  2as

This is the third equation of motion. From this equation, we can calculate the distance traveled s, if we know the initial and final velocity, without exactly knowing the time taken for the travel.

Example 1: A scooter starts from rest and achieves a final velocity of 20km/hour in a distance of 50m. Find the acceleration of the scooter.

Substituting  u = 0, v = 20 km/hour, s = 50m in the third equation of motion, we get

(20km/hour) ² = 2 x a x 50m

Converting all units of kilometer and hour to meter and seconds,

We get   a =  0.31 m/s²

Summary
Motion of bodies can be calculated from the three equations of motions. Quite often, more than one equation has to be considered for solving a certain problem. Thus given certain initial parameters such as initial velocity, acceleration, etc. one can determine completely the motion of a body by using these equations of motion.  

Next        Main        Previous