a particle is moving in a straight line with initial velocity U and uniform acceleration a. If the sum of the distances travelled in t to the power of t h space space space space space space end exponent a n d space left parenthesis t plus 1 right parenthesis to the power of t h end exponentseconds in 100cm , then its velocity after t seconds in cm/s is

Asked by Petulla Mishra | 4th Nov, 2015, 10:16: AM

Expert Answer:

We know that the distance travelled in nth second of uniformly accelerated motion is:
begin mathsize 14px style straight S subscript straight n equals straight u plus straight a over 2 left parenthesis 2 straight n minus 1 right parenthesis end style
where u is the initial velocity and a the uniform acceleration of the body.
In this case given that the sum of distances travelled in tth & (t+1)th second is 100 cm.
i.e.
S+ St+1 = 100
If the particle is moving with initial velocity U and uniform accerleration a, then the distance travelled by the particle in tth second can be written as:
begin mathsize 14px style straight S subscript straight t equals straight U plus straight a over 2 left parenthesis 2 straight t minus 1 right parenthesis space space space space... space left parenthesis 1 right parenthesis Distance space travelled space by space the space particle space in space left parenthesis straight t plus 1 right parenthesis to the power of th space second comma straight S subscript straight t plus 1 end subscript equals straight U plus straight a over 2 left parenthesis 2 left parenthesis straight t plus 1 right parenthesis minus 1 right parenthesis straight S subscript straight t plus 1 end subscript equals straight U plus straight a over 2 left parenthesis 2 straight t plus 1 right parenthesis space space space space space... space left parenthesis 2 right parenthesis end style
Given that St + St+1 = 100
Substituting eqn (1) and (2), we get,
begin mathsize 14px style straight U plus straight a over 2 left parenthesis 2 straight t minus 1 right parenthesis plus straight U plus straight a over 2 left parenthesis 2 straight t plus 1 right parenthesis equals 100 On space solving space we space get comma straight U plus at minus straight a over 2 plus straight U plus at plus straight a over 2 equals 100 2 straight U plus 2 at equals 100 rightwards double arrow straight U plus at space equals space 50 end style
According to the equation of motion, we know that v = u+at, where v is the final velocity, u is the initial velocity, a the acceleration and t the time.
So comparing the equation U+at =50 with the the equation of motion v= u+at, we get that the final velocity of the particle at t second = 50 cm/s.

Answered by Faiza Lambe | 4th Nov, 2015, 11:18: AM