Question

Derive third equation of motion
( v^2= u^2+2as).
​

Answer 1

The first two equations of motion are:

1. $\sf{v=u+at}$
2. $\sf{s=ut+\dfrac{1}{2} at^2}$

1st equation, $\sf{v=u+at}$ is available.

$\sf{v^2=(u+at)^2}$

$\sf{v^2=u^2+2aut+a^2t^2}$

$\sf{v^2=u^2+a(2ut+at^2)}$

2nd equation, $\sf{2s=2ut+at^2}$ is derived after multiplying sides by 2.

$\sf{v^2=u^2+a(2s)}$

$\boxed{\sf{\therefore v^2=u^2+2as}}$

Derivation of Equations:

1st eq: Let's derive $\sf{v=u+at}$.

By use of $\sf{a=\dfrac{v-u}{t} }$  [The Definition of Acceleration]

→ $\sf{a\times t=\dfrac{v-u}{t} \times t}$

→ $\sf{at=v-u}$

Flip sides!

→ $\sf{v-u=at}$

$\boxed{\sf{\therefore v=u+at}}$

2nd eq: The equation of motion $\sf{s=ut+\dfrac{1}{2}at^2 }$.

By use of the Velocity-Time Graph

In an initial acceleration, the graph will form a line.

• v is the final velocity
• u is the initial velocity
• t is time

The area under the graph consists of a right triangle and a rectangle.

And, the area under the graph is the displacement s.

• Right Triangle: $\sf{\dfrac{1}{2} \times t\times (v-u)}$
• Rectangle: $\sf{ut}$

The first equation, $\sf{v-u=at}$ into the triangle area.

→ $\sf{\dfrac{1}{2} at^2}$ is the area of a right triangle.

→ Therefore, the displacement is $\sf{ut+\dfrac{1}{2} at^2}$.

$\boxed{\sf{\therefore s=ut+\dfrac{1}{2}at^2}}$

Answer 2

The first two equations of motion are:

\sf{v=u+at}v=u+at

\sf{s=ut+\dfrac{1}{2} at^2}s=ut+

2

1

at

2

By use of 1

\sf{v^2=(u+at)^2}v

2

=(u+at)

2

\sf{v^2=u^2+2aut+a^2t^2}v

2

=u

2

+2aut+a

2

t

2

\sf{v^2=u^2+a(2ut+at^2)

By use of 2

\sf{v^2=u^2+a(2s)

\sf{\therefore{v^2=u^2+2as}}∴v

2

=u

2

+2as

For more:

Let's derive \sf{v=u+at}v=u+at , the first equation of motion.

By use of \sf{a=\dfrac{v-u}{t} }a=

t

v−u

, definition of accerlation

\sf{at=v-u}at=v−u

\sf{\therefore{v=u+at}}∴v=u+at

The second equation of motion \sf{s=u^2+\dfrac{1}{2}at^2 }s=u

2

+

2

1

at

2

.

Velocity-Time Graph

The area under the graph is \sf{u^2+\dfrac{1}{2}at^2 }u

2

+

2

1

at

2

which is the displacement.

Hope it will help you Mark me as brainlist