The acceleration-time graph of a particle moving in a straight line is as shown in figure. The velocity of the particle at time t = 0 is 2 m/s. The velocity after 2 seconds will be ?
(
)
options :-
(1) 6 m/s
(2) 4 m/s
(3) 2 m/s
(4) 8 m/s
please answer with full explanation
note: all lines in the graph are straight
()
options :-
(1) 6 m/s
(2) 4 m/s
(3) 2 m/s
(4) 8 m/s
please answer with full explanation
note: all lines in the graph are straight
1 Answer
I got
(1)
Explanation:
From Graph:
and
- from
t=0 tot=1\ s
dota=4\ ms^-3
Integrating both sides with time we getint\ dota\ dt=int\ 4\ dt
=>a=4t+C
whereC is a constant of integration to be calculated from initial conditions.
From Graph att=0 ,a=0
:.a=4t\ ms^-2 ........(1)
Rewriting this as
a=dotv=4t\ ms^-2
Integrating both sides with time we get
int\ dotv\ dt=int\ 4t\ dt
=>v=4t^2/2+C_1
=>v=2t^2+C_1
whereC_1 is a constant of integration to be calculated from initial conditions. Given, att=0 v=2\ ms^-1 . Inserting these values in above expression we get
=>2=2(0)^2+C_1
=>C_1=2
This gives us the expression for velocity as
v=2+2t^2 .......(2)
andv_(t=1)=4\ ms^-1 ...........(3) - from
t=1 tot=2\ s
dota=-4\ ms^-3
Integrating both sides with time we getint\ dota\ dt=int\ -4\ dt
=>a=-4t+C_2
whereC_2 is a constant of integration to be calculated from initial conditions att=1\ s .
From Graph att=1 ,a=4\ ms^-2 . Inserting in above expression we get
4=-4xx1+C_2
=>C_2=8
:.a=-4t+8\ ms^-2 ........(4)
Rewriting this as
a=dotv=-4t+8\ ms^-2
Integrating both sides with time we get
int\ dotv\ dt=int\ (-4t+8)\ dt
=>v=-4t^2/2+8t+C_3
=>v=-2t^2+8t+C_3
whereC_3 is a constant of integration to be calculated from calculated initial conditions att=1 v=4\ ms^-1 . Inserting these values in above expression we get
=>4=-2(1)^2+8xx1+C_3
=>C_3=-2
This gives us the expression for velocity as
v=-2t^2+8t-2 .......(3)
andv_(t=2)=(-2(2)^2+8xx2-2)
=>v_(t=2)=6\ ms^-1 ...........(6)