Question

A particle is moving in a straight line with initial velocity u and uniform acceleration a. If the sum of the distances travelled in the t second t+1 seconds is 100cm, then it's velocity after t second in cm/s is Please explain detailed answers for above questions.

Answer 1

Correct Question:

A particle is moving in straight line with initial velocity u and uniform acceleration a. If the sum of the distances travelled in the (t)th and (t+1)th seconds 100cm, then its velocity after t seconds in cm/s is

Given:

Initial velocity of particle= u

Acceleration of particle= a

Sum of distances travelled in the (t)th and (t+1)th seconds= 100 cm

To Find:

Velocity of given particle after t seconds in cm/s

Solution:

We know that,

• Distance travelled by a body in nth second is given by

$\pink{\boxed{\bf{S_{n}=u+\frac{1}{2}a(2n-1)}}}$

where,

u is the initial velocity

a is the acceleration

• According to first equation of motion for constant acceleration

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

Now,

Let the distance travelled by the given particle in t s be $\rm{S_{t}}$ and distance travelled in (t+1) s be $\rm{S_{t+1}}$

So,

$\longrightarrow\rm{S_{t}=u+\dfrac{1}{2}a(2t-1)}$

$\longrightarrow\rm{S_{t}=u+\dfrac{a(2t-1)}{2}}----(1)$

Also,

$\longrightarrow\rm{S_{t+1}=u+\dfrac{1}{2}a(2(t+1)-1)}$

$\longrightarrow\rm{S_{t+1}=u+\dfrac{1}{2}a(2t+2-1)}$

$\longrightarrow\rm{S_{t+1}=u+\dfrac{1}{2}a(2t+1)}$

$\longrightarrow\rm{S_{t+1}=u+\dfrac{a(2t+1)}{2}}----(2)$

Now,

According to question,

$\longrightarrow\rm{S_{t}+S_{t+1}=100}$

From (1) and (2), we get

$\longrightarrow\rm{u+\dfrac{a(2t-1)}{2}+u+\dfrac{a(2t+1)}{2}=100}$

$\longrightarrow\rm{u+u+\dfrac{a(2t-1)}{2}+\dfrac{a(2t+1)}{2}=100}$

$\longrightarrow\rm{2u+\dfrac{a}{2}\Big(2t-1+2t+1\Big)=100}$

$\longrightarrow\rm{2u+\dfrac{a}{\cancel{2}}\big(\cancel{4}t\big)=100}$

$\longrightarrow\rm{2u+2at=100}$

$\longrightarrow\rm{2(u+at)=100}$

$\longrightarrow\rm{u+at=\dfrac{\cancel{100}}{\cancel{2}}}$

$\longrightarrow\rm{u+at=50}------(3)$

Now,

Let the velocity of particle after t s be 'v' cm/s

So, on using first equation of motion, we get

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

From (3) ,we get

$\longrightarrow\rm\green{v=50\:cm/s}$

Hence, the velocity of given particle after t seconds  is 50 cm/s.