PHYSICS||KINENATICS||. que no.81

Asked by agrawalpulkit2004 | 14th Apr, 2020, 09:50: PM

Expert Answer:

A car starts from rest and moves with a uniform acceleration of α m/s2 along a straight line. It then retards uniformly at a rate of β m/s2 and stops. If 't' is the time elapsed since it stats moving and stops, find the average speed of the car.
 
Average speed = total distance/ time = Area under v-t graph curve/ time 
 
B e f o r e space t h e space r e t a r d a t i o n space s t a r t s comma space
E q u a t i o n space o f space l i n e space i s space y space equals space m x plus c
v space equals space u space plus space a t space
v 1 space equals space 0 plus alpha t 1
rightwards double arrow t 1 equals fraction numerator v 1 over denominator alpha end fraction space... space left parenthesis 1 right parenthesis

a f t e r space s o m e space p o i n t comma space c a r space s t a r t s space r e t a r d i n g space a n d space s t o p s
E q u a t i o n space o f space o t h e r space l i n e space i s comma space
0 space equals space v 1 plus beta left parenthesis t space minus space t 1 right parenthesis
T h u s comma space
v 1 space equals space beta left parenthesis t 1 space minus space t right parenthesis
rightwards double arrow fraction numerator v 1 over denominator beta end fraction space equals space t 1 space minus space t space... space left parenthesis 2 right parenthesis

S u b t r a c t i n g space e q u a t i o n space left parenthesis 2 right parenthesis space f r o m space left parenthesis 1 right parenthesis space w e space g e t comma space
fraction numerator v 1 over denominator alpha end fraction minus fraction numerator v 1 over denominator beta end fraction equals t 1 minus space t 1 space plus space t
v 1 open parentheses 1 over alpha minus 1 over beta close parentheses equals t

v 1 space equals open parentheses fraction numerator alpha beta over denominator beta minus alpha end fraction close parentheses t space space... space left parenthesis 3 right parenthesis space

A r e a space u n d e r space t h e space s h a d e d space r e g i o n space i s space g i v e n space a s comma
n e t space d i s p l a c e m e n t space o f space c a r space space equals space 1 half t cross times v 1 space
equals space 1 half cross times t cross times open parentheses fraction numerator alpha beta over denominator beta minus alpha end fraction close parentheses t space space... space left parenthesis space f r o m space left parenthesis 3 right parenthesis right parenthesis

T h u s space a v e r a g e space s p e e d space o f space c a r space equals space space n e t space d i s p l a c e m e n t space divided by space t i m e space equals space a r e a space u n d e r space t h e space c u r v e divided by t i m e space
equals space fraction numerator 1 half open parentheses fraction numerator alpha beta over denominator beta minus alpha end fraction close parentheses t squared over denominator t end fraction space equals 1 half open parentheses fraction numerator alpha beta over denominator beta minus alpha end fraction close parentheses t space equals space open square brackets fraction numerator alpha beta over denominator 2 left parenthesis beta minus alpha right parenthesis end fraction close square brackets t

Answered by Shiwani Sawant | 15th Apr, 2020, 02:19: PM