Question

A particle starts with initial velocity u and after some time t its velocity becomes v. The velocity at t/2
is

Answer 1

Given:

Initial velocity of particle= u

Final velocity of particle= v

Time taken= t

To Find:

The velocity at time t/2

Solution:

Let the velocity of particle at time t/2 be v'

We know that,

• According to first equation of motion for constant acceleration

$\pink{\boxed{\bf{v=u+at}}}$

where,

• v is the final velocity
• u is the initial velocity
• a is acceleration
• t is time taken

For the given particle when it has travelled for time 't'

On using first equation of motion, we get

$\longrightarrow\mathrm{v=u+at}$

$\longrightarrow\mathrm{v-u=at}$

$\longrightarrow\mathrm{at=v-u}$

$\longrightarrow\mathrm{a=\dfrac{v-u}{t}}------(1)$

Now, when it travelled for time t/2

On using first equation of motion, we get

$\longrightarrow\mathrm{v'=u+a\bigg(\dfrac{t}{2}\bigg)}$

From (1), we get

$\longrightarrow\mathrm{v'=u+\dfrac{v-u}{\cancel{t}}\bigg(\dfrac{\cancel{t}}{2}\bigg)}$

$\longrightarrow\mathrm{v'=u+\dfrac{v-u}{2}}$

$\longrightarrow\mathrm{v'=\dfrac{2u+v-u}{2}}$

$\longrightarrow\mathrm{\green{v'=\dfrac{u+v}{2}}}$