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` tell me the fourth equation of motion and meaning of each symbol occurring in it...if possible also tell the consequent equations of motion...`

6 years ago

Dear Indrayani

- v= u + at
- s =( u + v ) t /2
- s = ut + at
^{2}/2- v
^{2}= u^{2}+ 2as

The following are thevariablesused in these equations:

- t is the length of the interval of time and measured in seconds.
- u is the initial speed of the object or the speed at the beginning of the time interval, measured in ms
^{-1}.- v is the final speed of the object or the speed at the end of the time interval, measured in ms
^{-1}.- a is the acceleration of the object within the time interval t measured in ms
^{-2}.- s is the distance covered or the displacement of the object during the time interval t, measured in metres.
Derivation of Newton's Equations of Motion:

Derivation of First and second equations of motion:

We know that,Velocity is the Change in displacement / Change in time,

v = ds/dt = s/t andAcceleration is the change in velocity / change in time,

a = dv/dt = (v-u)/t

Or the acceleration is the change in speed per unit time, so:

a = (v-u) /t

or at = v-u

or v= u + at and it is thefirst Equation of Motion.

We also know that the average speed during the time interval t is equal to displacement per unit time, so:

(u + v)/2 = s / t since Average speed = (u+v)/2

s / t = (u + v)/2

ors = ( u + v ) t /2and this is thesecond equation of motion.

Derivation of third and Fourth Equations of Motion:Derivation of third equation of motion

The third equation of motion can be derived by substituting the value of v from the first equation into the second equation.

First equation isv= u +at

And the second equation is s = (u+v)t /2

Or s=[u + (u+at)] t /2 =[2u+at]t /2 = [2ut+ at2] / 2 =ut + at2/2

Or s=ut + at2/2 and this is thethird equation of motion.

Derivation of fourth equation of motion

To derive the fourth equation of motion, first we have to consider the equation for acceleration and then to rearrange it.

Accelerationa = (v-u) /t

Or at = (v-u)

Ort = (v-u) /a

Next step is to substitute the equation for the time t into the second equation of motion s=ut + at2/2 to obtain:

s = u[(v-u)/a] + a [(v-u)/a]^{2}/2

or s = u(v-u)/a + (v-u)^{2}/2a

or s =[2u+v-u](v-u)/2a

or s = (u+v)(v-u)/2a = (v+u)(v-u)/2a

ors= (v^{2}-u^{2})/2a

or2as = v^{2}–u^{2}^{}

orv^{2}–u^{2}= 2as

or v^{2}= u^{2}+ 2as and thisequation of motioncan be used to find the final velocity or the distance travelled if the other values are given.All the best.AKASH GOYALAskiitiansExpert-IIT Delhi

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6 years ago

S=(u+v)t/2Derivation-We know that a=(v-u)/t and s=ut+(at^2)/2S=ut+((v-u)/t)t^2S=ut+(v-u)t/2Take t as commonS=t[2u+v-u]/2S=t(v+u)/2 Proved

6 months ago

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