Question

The velocity versus time graph of two particles
moving along x-axis varies as shown in the
following two plots. Then

(1) Maximum separation between two particle is 2m
(2) Maximum separation between the two particles
is 3 m
(3) Maximum separation occurs at t = 2 s
(4) Maximum separation occurs at t = 3 s​

Answer 1

The maximum separation occurs at t = 3 seconds and the maximum separation between two particles is 3 m

Options (2) and (4) are correct.

Explanation:

Area under the velocity time graph gives the displacement

Case I

If t < 2 seconds

For the first particle the displacement in time t

$s_1=\frac{1}{2}\times 2\times t$

$\implies s_1=t$

For the second particle the displacement in time t

$s_2=\frac{1}{2}\times 4\times t$

$\implies s_2=2t$

Separation between the two particles

$=s_2-s_1$

$=2t-t$

$=t$

The separation is linearly dependent on time t

Thus, when time t is maximum the separation will be maximum

The maximum time will be here t = 2 seconds

Case II

t ≥ 2 seconds

For the first particle the displacement in time t

$s_1'=\frac{1}{2}\times 2\times 2+2(t-2)$

$\implies s_1'=2t-2$

For the second particle the displacement in time t

$s_2'=\frac{1}{2}\times 4\times 2+(\frac{1}{2}\times 4\times 2-\frac{1}{2}\times (-2t+8)(4-t))$     (Note the relation between the v and t line between 2 to 4 seconds is v = -2t +8)

$\implies s_2'=4+4-(4-t)^2$

$\implies s_2'=8-(4-t)^2$

Separation between the two particles

$=s_2'-s_1'$

$=8-(4-t)^2-(2t-2)$

$=8-(16-8t+t^2)-2t+2$

$=-8+8t-t^2-2t+2$

$=-6+6t-t^2$

$=-(t^2-6t+6)$

$=-(t^2-2\times 3t+3^2-3^2+6)$

$=-(t-3)^2+3$

$=3-(t-3)^2$

The maximum value of separation occurs when t = 3 seconds

Therefore, maximum separation when t ≥ 2 is 3 m

Therefore, the maximum separation occurs at t = 3 seconds and the maximum separation between two particles is 3 m

Hope this answer is helpful.

Know More:

Q: Acceleration time graph for particle moving along x axis from rest is shown in figure if initial velocity of particle is -5m/s then velocity at time t=8 sec is:

Click Here: https://brainly.in/question/4537939

Q: The displacement-time graph for two particles A and B are straight lines inclined at angles of 30° and 45° with the time axis. The ratio of the velocities of A:B is:

Click Here: https://brainly.in/question/1323482