# Motion in Straight Line

## Third Equation of Motion:

The third equation of motion is derived by substituting the value of time (t) from first equation of motion.

We know from first equation of motion, `v=u+at`

`=>v-u=at`

`=>at=v-u`

`=>t=(v-u)/a` -----(v)

We know that the second equation of motion is, `s=ut+1/2at^2`

By substituting the value of `t` from euqation (v) we get

`s=u((v-u)/a)+1/2a((v-u)/a)^2`

`=>s=uxx(v-u)/a+1/2a((v-u)^2)/a^2`

`=>s=(u(v-u))/a +(axx(v-u)^2)/(2xxaxxa)`

`=>s=(uv-u^2)/a + ((v-u)^2)/(2a)`

`=>s=(2(uv-u^2)+(v-u)^2)/(2a)`

`=>2as=2uv-2u^2+v^2+u^2-2uv`

`=>2as=-2u^2+v^2+u^2`

`=>2as=-u^2+v^2`

`=>2as+u^2=v^2`

`=>v^2=u^2+2as` ---(vi)

This is called the Third equation of motion.

Note: You can also use v_{0} in place of u for initial velocity.

### Free Fall

When an object is released near the surface of the Earth, it is accelerated downward under the influence of force of gravity. If we neglect air resistance, the object is said to be in free fall.

The object is released from y = 0 and hence v_{0} = 0

In this case, equations of motion can be written as follows:

v = 0 – gt = -9.8 t ms^{-1}

y = 0 – ½ gt^{2} = -4.9 t^{2} m

v^{2} = 0 – 2gy = -19.6y m^{2} s^{-2}

#### Galileo's Law of Odd Numbers:

The distance traversed during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity (1: 3: 5: 7 ………..)

**Proof:** Let us divide the time interval of motion of an object under free fall into many equal time intervals τ (pronounced as tau). After that, we need to find the distances traveresed during successive intervals of time. Distance can be calculated using following formula:

`y=-1/2gt^2`

Different time intervals can be taken as 0, τ 2τ, 3τ and so on. On calculation, it is seen that distances show the simple ratio of 1: 3: 5: 7 ……………..

### Reaction Time:

When something is dropped from a certain height and it drops udner free fall, you may need to catch it in time to prevent its fall on the earth. The time needed to capture the object is called reaction time. The distance travelled (d) and reaction time (t_{r}) can be given by following equation:

`d=-1/2g\t_r^2`

Or, `t_r=sqrt((2d)/g)` s