Question 55: Motion in 1D

Question

The velocity $$-$$ displacement graph of a particle is shown in the figure.The acceleration $$-$$ displacement graph of the same particle is represented by :

Accepted Answer

The slope of the given v versus x graph is $$m =  - {{{v_0}} \over {{x_0}}}$$ and intercept is c = + v
0
. Hence, v varies with x as
$$v =  - \left( {{{{v_0}} \over {{x_0}}}} \right)x + {v_0}$$ ...... (1)
where v
0
and x
0
are constants of motion. Differentiating with respect to time t, we have
$${{dv} \over {dt}} =  - \left( {{{{v_0}} \over {{x_0}}}} \right){{dx} \over {dt}}$$
or $$a =  - \left( {{{{v_0}} \over {{x_0}}}} \right)v$$ ........ (2)
Using Eq. (1) in Eq. (2), we get
$$a =  - \left( {{{{v_0}} \over {{x_0}}}} \right)\left( { - {{{v_0}} \over {{x_0}}}x + {v_0}} \right)$$
$$a = {\left( {{{{v_0}} \over {{x_0}}}} \right)^2}x - {{v_0^2} \over {{x_0}}}$$
Thus the graph of a versus x is a straight line having a positive slope = $${\left( {{{{v_0}} \over {{x_0}}}} \right)^2}$$ and negative intercept = $$ - {{v_0^2} \over {{x_0}}}$$. Hence the correct choice is (a).

Answer

Answer

Answer

Answer

Motion in 1D

The velocity $- displacement graph of a particle is shown in the figure.The acceleration -$ displacement graph of the same particle is represented by :

Choose the correct answer: