DERIVE

V = U+AT

V^2 -U^2= 2AS

g =GMM/R^2

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Solution

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 that, 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”. We know that

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

g=GM/R²

Weight of an object on Earth = Gravitational force exerted on an object by the Earth.
Let us consider a body having mass "m".
mg=G⋅mM/R ²
where,
g=acceleration due to gravity,
G=Gravitational constant,
M=Mass of the Earth,
and R=radius of the Earth.
Weight of the body (on Earth)=
mass(m)⋅gravitational acceleration(g)
Cancel out the m on both sides to get:
g=G⋅M/R ²=GM/R ²

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How the following equations are dimensionally correct -

[F]=G Mm/r^2

And

V^2 - u^2=2as

p (u) = u^{2} + 3 u + 4, q (u) = u^{2} + u - 12,

r (u) = u - 2

p (u) q (u) r (u)

∣ ∣ ∣ ∣ ∣ \begin{matrix} y z - x^{2} & z x - y^{2} & x y - z^{2} x z - y^{2} & x y - z^{2} & y z - x^{2} x y - z^{2} & y z - x^{2} & z x - y^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} r^{2} & u^{2} & u^{2} u^{2} & r^{2} & u^{2} u^{2} & u^{2} & r^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

u = f (r)

r^{2} = x^{2} + y^{2} + z^{2}

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = f^{''} (r) + \frac{2}{r} f (r)

u = f (r), r^{2} = x^{2} + y^{2}

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = f (r) + \frac{2}{r} f^{'} (r)

u = f (r), r^{2} = x^{2} + y^{2} + z^{2}

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = f (r) + \frac{1}{r} f^{'} (r)

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