Question

How are 3 equations of Motion derive for class 9​

Answer 1

Equations of Motions

In case of uniform acceleration, there are three equations of motion which are also known as the laws of constant acceleration. Hence, these equations are used to derive the components like displacement(s), velocity (initial and final), time(t) and acceleration(a). Therefore they can only be applied when acceleration is constant and motion is a straight line. The three equations are,

v = u + at

v² = u² + 2as

s = ut + ½at²

where, s = displacement; u = initial velocity; v = final velocity; a = acceleration; t = time of motion. These equations are referred as SUVAT equations where SUVAT stands for displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (T)

Equations of Motion

Derivation of the Equations of Motion

v = u + at

Let us begin with the first equation, v=u+at. This equation only talks about the acceleration, time, the initial and the final velocity. Let us assume a body that has a mass “m” and initial velocity “u”. Let after time “t” its final velocity becomes “v” due to uniform acceleration “a”. Now we know that:

Acceleration = Change in velocity/Time Taken

Therefore, Acceleration = (Final Velocity-Initial Velocity) / Time Taken

Hence, a = v-u /t or at = v-u

Therefore, we have: v = u + at

v² = u² + 2as

We have, v = u + at. Hence, we can write t = (v-u)/a

Also, we know ,Distance = average velocity × Time

Therefore, for constant acceleration we can write: Average velocity = (final velocity + initial velocty)/2 = (v+u)/2

Hence, Distance (s) = [(v+u)/2] × [(v-u)/a]

or s = (v² – u²)/2a

or 2as = v² – u²

or v² = u² + 2as

s = ut + ½at²

Let the distance be “s”.

Distance = Average velocity × Time. Also, Average velocity = (u+v)/2

Therefore, Distance (s) = (u+v)/2 × t

Also, from v = u + at, we have:

s = (u+u+at)/2 × t = (2u+at)/2 × t

s = (2ut+at²)/2 = 2ut/2 + at²/2

or s = ut +½ at²

hope it helps...............

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Answer 2

The 3 equations of motion are:-

•v=u+at

•s=ut+1/2at²

•v²-u²=2as

Where:-

•v is final velocity of the body

•u is initial velocity of the body

•s is distance/displacement of the body

•a is acceleration of the body

•t is time taken

Note:-The three equations of motion are only valid when a body is moving with uniform acceleration.

We can derive the 3 equations of motion by 2 methods.

•Graphical method

•Non-graphical method

Here,we shall derive the equations of motion by Non-graphical method.

Derivation of 1st equation of motion:-

=>Acceleration=Change in velocity/Time

=>a= v-u/t

=>v=u+at -------(1)

Derivation of 2nd equation of motion:-

For uniformly accelerated motion,

Vav=u+v/2

Also,Displacement= (Average Velocity)×(Time)

=>s=(u+v/2)×t

=>s=(u+u+at/2)t (since,v=u+at)

=>s=ut+1/2at² -----(2)

Derivation of 3rd equation of motion:-

From(1), t=v-u/a

Substituting the value of t in the second equation of motion,

=>s=u(v-u/a)+1/2a(v-u/a)²

=>s= uv-u²/a+1/2a(v²+u²-2uv)

=>s=2uv-2u²+v²+u²-2uv/2a

=>s=v²-u²/2a

=>v²-u²=2as -----(3)