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 ?

(useruploads.socratic.org)

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
Feb 17, 2018

I got
(1)

Explanation:

From Graph: dota=4\ ms^-3 from t=0 to t=1\ s
and dota=-4\ ms^-3 from t=1 to t=2\ s

  1. from t=0 to t=1\ s
    dota=4\ ms^-3
    Integrating both sides with time we get

    int\ dota\ dt=int\ 4\ dt
    =>a=4t+C
    where C is a constant of integration to be calculated from initial conditions.
    From Graph at t=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
    where C_1 is a constant of integration to be calculated from initial conditions. Given, at t=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)
    and v_(t=1)=4\ ms^-1 ...........(3)

  2. from t=1 to t=2\ s
    dota=-4\ ms^-3
    Integrating both sides with time we get

    int\ dota\ dt=int\ -4\ dt
    =>a=-4t+C_2
    where C_2 is a constant of integration to be calculated from initial conditions at t=1\ s.
    From Graph at t=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
    where C_3 is a constant of integration to be calculated from calculated initial conditions at t=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)
    and v_(t=2)=(-2(2)^2+8xx2-2)
    =>v_(t=2)=6\ ms^-1 ...........(6)