Question

The magnitude of the velocity of the body:

$(a)$ Increases between points O and A
$(b)$Increases between point A and B
$(c)$ Decreases between point A and B
$(d)$ Is zero throughout

Answer 1

Hint: In this question use the concept that velocity is defined as the rate change of position with respect to time that is $v = \dfrac{{dx}}{{dt}}$. So it is clear that the slope of the displacement time graph will eventually give the velocity. Check for the portions in the graph where the slope is increasing, decreasing or is constant. This will help get the right answer for this problem statement.

Complete step-by-step answer:
We all know the relation between the speed, displacement and time.
As the velocity of any particle is the ratio of the displacement to time taken by the particle.
As we all know that the speed is a scalar quantity (i.e. only magnitude no direction), but velocity is a vector quantity it possesses magnitude as well as direction.
The direction is dependent on the movement of the particle but its magnitude is always equal to the speed of the particle.
Now in a given displacement time graph up to point A i.e. from point O to point A, the slope is decreasing i.e. the rate of change of displacement with respect to the time is decreasing so the particle velocity is also decreasing (I.e. whenever a particle is thrown from the rest it speed starts decreasing so the magnitude of the slope also decreases in nature after reaching the highest point, when the particle starts coming down its velocity starts increasing so the magnitude of the slope also increases).
As in displacement time graph the velocity is nothing but the slope of the graph i.e. rate of change of displacement with respect to time.
$\Rightarrow v = \dfrac{{dx}}{{dt}}$, where v = velocity, x = displacement and t = time.
So the magnitude of the velocity is
$\Rightarrow \left| v \right| = \left| {\dfrac{{dx}}{{dt}}} \right|$
Now it is clear from the figure that the magnitude of the slope is decreasing from point O to point A and it is again starting increasing from point A to point B.
So this is the required answer.
Hence option (B) is the correct answer.

Note – The slope of a straight line is defined as the angle it makes with the positive direction of the x-axis in anticlockwise sense. Since $v = \dfrac{{dx}}{{dt}}$ thus the slope has a relationship with the velocity. Now as acceleration that is $a = \dfrac{{dv}}{{dt}}$ so the slope of a velocity-time curve will give the value of acceleration. At the point A on the curve the slope is 0 as the tangent at point ‘A’ will be parallel to the x-axis and thus the velocity will be zero as well.